| L(s) = 1 | + 4·2-s + 4·4-s + 6·5-s − 6·7-s − 4·8-s − 7·9-s + 24·10-s + 4·11-s − 24·14-s − 8·16-s − 4·17-s − 28·18-s + 2·19-s + 24·20-s + 16·22-s − 12·23-s + 8·25-s + 2·27-s − 24·28-s − 8·29-s − 14·31-s − 16·34-s − 36·35-s − 28·36-s + 12·37-s + 8·38-s − 24·40-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2·4-s + 2.68·5-s − 2.26·7-s − 1.41·8-s − 7/3·9-s + 7.58·10-s + 1.20·11-s − 6.41·14-s − 2·16-s − 0.970·17-s − 6.59·18-s + 0.458·19-s + 5.36·20-s + 3.41·22-s − 2.50·23-s + 8/5·25-s + 0.384·27-s − 4.53·28-s − 1.48·29-s − 2.51·31-s − 2.74·34-s − 6.08·35-s − 4.66·36-s + 1.97·37-s + 1.29·38-s − 3.79·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.71842724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.71842724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( ( 1 + T )^{6} \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + 3 p^{2} T^{2} - 7 p^{2} T^{3} + 7 p^{3} T^{4} - 3 p^{5} T^{5} + 145 T^{6} - 3 p^{6} T^{7} + 7 p^{5} T^{8} - 7 p^{5} T^{9} + 3 p^{6} T^{10} - p^{7} T^{11} + p^{6} T^{12} \) |
| 3 | \( 1 + 7 T^{2} - 2 T^{3} + 28 T^{4} + 2 T^{5} + 100 T^{6} + 2 p T^{7} + 28 p^{2} T^{8} - 2 p^{3} T^{9} + 7 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 - 6 T + 28 T^{2} - 4 p^{2} T^{3} + 333 T^{4} - 878 T^{5} + 2121 T^{6} - 878 p T^{7} + 333 p^{2} T^{8} - 4 p^{5} T^{9} + 28 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 4 T + 49 T^{2} - 204 T^{3} + 1152 T^{4} - 4240 T^{5} + 16164 T^{6} - 4240 p T^{7} + 1152 p^{2} T^{8} - 204 p^{3} T^{9} + 49 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 17 | \( 1 + 4 T + 81 T^{2} + 280 T^{3} + 3074 T^{4} + 8724 T^{5} + 67033 T^{6} + 8724 p T^{7} + 3074 p^{2} T^{8} + 280 p^{3} T^{9} + 81 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 - 2 T + 87 T^{2} - 10 p T^{3} + 3512 T^{4} - 7188 T^{5} + 84124 T^{6} - 7188 p T^{7} + 3512 p^{2} T^{8} - 10 p^{4} T^{9} + 87 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 12 T + 118 T^{2} + 772 T^{3} + 4479 T^{4} + 1000 p T^{5} + 111732 T^{6} + 1000 p^{2} T^{7} + 4479 p^{2} T^{8} + 772 p^{3} T^{9} + 118 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 + 8 T + 130 T^{2} + 594 T^{3} + 6099 T^{4} + 18374 T^{5} + 187029 T^{6} + 18374 p T^{7} + 6099 p^{2} T^{8} + 594 p^{3} T^{9} + 130 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 + 14 T + 216 T^{2} + 1972 T^{3} + 17840 T^{4} + 117486 T^{5} + 749554 T^{6} + 117486 p T^{7} + 17840 p^{2} T^{8} + 1972 p^{3} T^{9} + 216 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 - 12 T + 135 T^{2} - 1176 T^{3} + 10575 T^{4} - 70860 T^{5} + 472322 T^{6} - 70860 p T^{7} + 10575 p^{2} T^{8} - 1176 p^{3} T^{9} + 135 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( 1 - 28 T + 503 T^{2} - 6376 T^{3} + 64135 T^{4} - 528684 T^{5} + 3676082 T^{6} - 528684 p T^{7} + 64135 p^{2} T^{8} - 6376 p^{3} T^{9} + 503 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 - 2 T + 149 T^{2} - 340 T^{3} + 11408 T^{4} - 21594 T^{5} + 590652 T^{6} - 21594 p T^{7} + 11408 p^{2} T^{8} - 340 p^{3} T^{9} + 149 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 14 T + 244 T^{2} - 2532 T^{3} + 25548 T^{4} - 209078 T^{5} + 1534242 T^{6} - 209078 p T^{7} + 25548 p^{2} T^{8} - 2532 p^{3} T^{9} + 244 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 + 22 T + 409 T^{2} + 5130 T^{3} + 58074 T^{4} + 513982 T^{5} + 4153497 T^{6} + 513982 p T^{7} + 58074 p^{2} T^{8} + 5130 p^{3} T^{9} + 409 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 + 2 T + 192 T^{2} + 380 T^{3} + 20828 T^{4} + 39642 T^{5} + 1464142 T^{6} + 39642 p T^{7} + 20828 p^{2} T^{8} + 380 p^{3} T^{9} + 192 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 + 14 T + 279 T^{2} + 2854 T^{3} + 32699 T^{4} + 263412 T^{5} + 2369290 T^{6} + 263412 p T^{7} + 32699 p^{2} T^{8} + 2854 p^{3} T^{9} + 279 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 - 24 T + 496 T^{2} - 6632 T^{3} + 80672 T^{4} - 775096 T^{5} + 6983258 T^{6} - 775096 p T^{7} + 80672 p^{2} T^{8} - 6632 p^{3} T^{9} + 496 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( 1 - 4 T + 358 T^{2} - 1164 T^{3} + 57375 T^{4} - 149608 T^{5} + 5246868 T^{6} - 149608 p T^{7} + 57375 p^{2} T^{8} - 1164 p^{3} T^{9} + 358 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) |
| 73 | \( 1 - 36 T + 919 T^{2} - 15928 T^{3} + 225311 T^{4} - 2516900 T^{5} + 23841506 T^{6} - 2516900 p T^{7} + 225311 p^{2} T^{8} - 15928 p^{3} T^{9} + 919 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 + 28 T + 686 T^{2} + 11252 T^{3} + 159023 T^{4} + 1794648 T^{5} + 17548548 T^{6} + 1794648 p T^{7} + 159023 p^{2} T^{8} + 11252 p^{3} T^{9} + 686 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 26 T + 684 T^{2} - 10460 T^{3} + 156752 T^{4} - 1683426 T^{5} + 17728366 T^{6} - 1683426 p T^{7} + 156752 p^{2} T^{8} - 10460 p^{3} T^{9} + 684 p^{4} T^{10} - 26 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( 1 - 42 T + 1087 T^{2} - 20330 T^{3} + 304908 T^{4} - 3736348 T^{5} + 38506140 T^{6} - 3736348 p T^{7} + 304908 p^{2} T^{8} - 20330 p^{3} T^{9} + 1087 p^{4} T^{10} - 42 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( 1 - 24 T + 7 p T^{2} - 10706 T^{3} + 174956 T^{4} - 2000590 T^{5} + 22996700 T^{6} - 2000590 p T^{7} + 174956 p^{2} T^{8} - 10706 p^{3} T^{9} + 7 p^{5} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.38214148681914448056464508528, −4.97198756547918707475596748080, −4.69861865804128786993380492295, −4.65836224289889344319540419431, −4.48231689601386507710951539574, −4.45534354027615169845676364917, −4.27626356369899245923873405634, −3.87788110178110834829326265764, −3.80119307390013222814968547935, −3.64273863894630003301110610144, −3.57408339232078730579409503325, −3.54013023422947938698569305173, −3.52586849597655241413870142372, −3.02102346912804160553227154322, −2.73130567511854820079895814731, −2.65388351025485967126480838234, −2.40082592635421042784848442429, −2.21897521597078647434448127446, −2.21168817826102840035626114450, −1.87565542996892409661334242786, −1.84079636640731647897466443701, −1.39599219755724444335375520479, −0.76277877609520481120464651042, −0.55495468338389605108456161863, −0.40841543941366507644304443617,
0.40841543941366507644304443617, 0.55495468338389605108456161863, 0.76277877609520481120464651042, 1.39599219755724444335375520479, 1.84079636640731647897466443701, 1.87565542996892409661334242786, 2.21168817826102840035626114450, 2.21897521597078647434448127446, 2.40082592635421042784848442429, 2.65388351025485967126480838234, 2.73130567511854820079895814731, 3.02102346912804160553227154322, 3.52586849597655241413870142372, 3.54013023422947938698569305173, 3.57408339232078730579409503325, 3.64273863894630003301110610144, 3.80119307390013222814968547935, 3.87788110178110834829326265764, 4.27626356369899245923873405634, 4.45534354027615169845676364917, 4.48231689601386507710951539574, 4.65836224289889344319540419431, 4.69861865804128786993380492295, 4.97198756547918707475596748080, 5.38214148681914448056464508528
Plot not available for L-functions of degree greater than 10.
