| L(s) = 1 | + 2·3-s + 4·5-s + 4·7-s + 3·9-s + 2·11-s − 4·13-s + 8·15-s + 2·17-s − 8·19-s + 8·21-s + 8·23-s + 8·25-s + 4·27-s − 2·29-s + 14·31-s + 4·33-s + 16·35-s + 2·37-s − 8·39-s − 2·41-s − 2·43-s + 12·45-s + 14·47-s + 4·49-s + 4·51-s + 16·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 1.51·7-s + 9-s + 0.603·11-s − 1.10·13-s + 2.06·15-s + 0.485·17-s − 1.83·19-s + 1.74·21-s + 1.66·23-s + 8/5·25-s + 0.769·27-s − 0.371·29-s + 2.51·31-s + 0.696·33-s + 2.70·35-s + 0.328·37-s − 1.28·39-s − 0.312·41-s − 0.304·43-s + 1.78·45-s + 2.04·47-s + 4/7·49-s + 0.560·51-s + 2.19·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61968384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61968384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.27521901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.27521901\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 97 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 260 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 156 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.019690782288763153150375085592, −7.81576665134836320236776472796, −7.16764314068692083628364912714, −7.14501780797635461579337103010, −6.72036738419521912365482738916, −6.29259007260656721079337129405, −5.93337542072038070496137656075, −5.63701147664871858907468128271, −4.97488482291146378268972080022, −4.94091675879303061206054119395, −4.61580994302979388053609125243, −4.04232631874472159624597874778, −3.88454896460230139006244227495, −3.06458272583166774105557038123, −2.69202466060637207128228355439, −2.43455857917051098388698990518, −1.94740801650735557710013292647, −1.87889963333742783881733445289, −0.986872332338873984203146283191, −0.941376433966038401525197662541,
0.941376433966038401525197662541, 0.986872332338873984203146283191, 1.87889963333742783881733445289, 1.94740801650735557710013292647, 2.43455857917051098388698990518, 2.69202466060637207128228355439, 3.06458272583166774105557038123, 3.88454896460230139006244227495, 4.04232631874472159624597874778, 4.61580994302979388053609125243, 4.94091675879303061206054119395, 4.97488482291146378268972080022, 5.63701147664871858907468128271, 5.93337542072038070496137656075, 6.29259007260656721079337129405, 6.72036738419521912365482738916, 7.14501780797635461579337103010, 7.16764314068692083628364912714, 7.81576665134836320236776472796, 8.019690782288763153150375085592
