| L(s) = 1 | + 3·2-s + 6·4-s + 5·5-s + 10·8-s + 15·10-s + 11-s + 2·13-s + 15·16-s + 4·17-s + 3·19-s + 30·20-s + 3·22-s + 7·23-s + 6·25-s + 6·26-s + 5·29-s + 14·31-s + 21·32-s + 12·34-s + 9·37-s + 9·38-s + 50·40-s + 12·41-s − 18·43-s + 6·44-s + 21·46-s − 3·47-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 2.23·5-s + 3.53·8-s + 4.74·10-s + 0.301·11-s + 0.554·13-s + 15/4·16-s + 0.970·17-s + 0.688·19-s + 6.70·20-s + 0.639·22-s + 1.45·23-s + 6/5·25-s + 1.17·26-s + 0.928·29-s + 2.51·31-s + 3.71·32-s + 2.05·34-s + 1.47·37-s + 1.45·38-s + 7.90·40-s + 1.87·41-s − 2.74·43-s + 0.904·44-s + 3.09·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(71.31957313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(71.31957313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 - p T + 19 T^{2} - 47 T^{3} + 19 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 7 T^{2} - 5 p T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 36 T^{2} - 49 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 7 T^{2} + 32 T^{3} + 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 107 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 7 T + 73 T^{2} - 319 T^{3} + 73 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 5 T + 55 T^{2} - 323 T^{3} + 55 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 138 T^{2} - 841 T^{3} + 138 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 102 T^{2} - 593 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 162 T^{2} - 1011 T^{3} + 162 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 210 T^{2} + 1549 T^{3} + 210 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 117 T^{2} + 309 T^{3} + 117 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 963 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 76 T^{2} + 11 p T^{3} + 76 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 4 T + 48 T^{2} - 229 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 5 T + 143 T^{2} + 521 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 25 T + 371 T^{2} - 3601 T^{3} + 371 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 7 T + 93 T^{2} + 335 T^{3} + 93 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 244 T^{2} + 1235 T^{3} + 244 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 9 T + 261 T^{2} - 1539 T^{3} + 261 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 28 T + 527 T^{2} - 5968 T^{3} + 527 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.81625118571674802260672401455, −6.33973043515108301056904068761, −6.32824886567540477696294813782, −6.32470095969015133436823761190, −5.82912763764518935651020368565, −5.82595567193704796624620922789, −5.66700187025915411257810484816, −5.11911490688705541065170348480, −5.00220219356206958648617393374, −4.97183296848503174212998539024, −4.56180873717483618705632949136, −4.47689899278778925959139341611, −4.17823050235400412435662472541, −3.65264642136979553333953062348, −3.42042728774178932367937514894, −3.35041172618748319932678764850, −2.93866985580743386660096913123, −2.79055364976688948434794313895, −2.58970415754928833306364726425, −2.02642829157449180340941009942, −1.96610868010291464893925229364, −1.73684441441446196309732902466, −1.11108933339772519787956192050, −1.02851876132500676045080785282, −0.75683427949782313319056661208,
0.75683427949782313319056661208, 1.02851876132500676045080785282, 1.11108933339772519787956192050, 1.73684441441446196309732902466, 1.96610868010291464893925229364, 2.02642829157449180340941009942, 2.58970415754928833306364726425, 2.79055364976688948434794313895, 2.93866985580743386660096913123, 3.35041172618748319932678764850, 3.42042728774178932367937514894, 3.65264642136979553333953062348, 4.17823050235400412435662472541, 4.47689899278778925959139341611, 4.56180873717483618705632949136, 4.97183296848503174212998539024, 5.00220219356206958648617393374, 5.11911490688705541065170348480, 5.66700187025915411257810484816, 5.82595567193704796624620922789, 5.82912763764518935651020368565, 6.32470095969015133436823761190, 6.32824886567540477696294813782, 6.33973043515108301056904068761, 6.81625118571674802260672401455
