| L(s) = 1 | − 6·3-s + 2·7-s + 27·9-s − 74·11-s + 98·13-s − 78·17-s + 80·19-s − 12·21-s − 40·23-s − 108·27-s + 50·29-s + 12·31-s + 444·33-s − 34·37-s − 588·39-s + 344·41-s − 216·43-s − 876·47-s + 478·49-s + 468·51-s − 634·53-s − 480·57-s − 666·59-s + 244·61-s + 54·63-s + 980·67-s + 240·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.107·7-s + 9-s − 2.02·11-s + 2.09·13-s − 1.11·17-s + 0.965·19-s − 0.124·21-s − 0.362·23-s − 0.769·27-s + 0.320·29-s + 0.0695·31-s + 2.34·33-s − 0.151·37-s − 2.41·39-s + 1.31·41-s − 0.766·43-s − 2.71·47-s + 1.39·49-s + 1.28·51-s − 1.64·53-s − 1.11·57-s − 1.46·59-s + 0.512·61-s + 0.107·63-s + 1.78·67-s + 0.418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 474 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 74 T + 3902 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 98 T + 6666 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 78 T + 8122 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 80 T + 7062 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 40 T + 16478 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 50 T + 43082 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 17822 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 34 T + 20970 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 876 T + 374206 T^{2} + 876 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 634 T + 398114 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 666 T + 211918 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 980 T + 692502 T^{2} - 980 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 308 T + 553262 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1412 T + 1090194 T^{2} + 1412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1052 T + 1237470 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 248 T + 1125926 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 684 T + 1229686 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1840 T + 2539650 T^{2} + 1840 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.134813727695630941301129936501, −8.727062162832495033338205550249, −8.188690276151103822548954148946, −8.095190877714211151420413806313, −7.42297558383710381328696778612, −7.16589876695208150011388504491, −6.35340026426278654169893536516, −6.35017905904169281722660168873, −5.72686912879348589735299017903, −5.49027878289581855034176344125, −4.80012424507694010567812018706, −4.69043599042521821225027955883, −3.99947599999390154890199283145, −3.40392273609469938900847431542, −2.92388713813338218541071789963, −2.27156924008469058727652243188, −1.47398797015446986921498309019, −1.10412142721995947334974773280, 0, 0,
1.10412142721995947334974773280, 1.47398797015446986921498309019, 2.27156924008469058727652243188, 2.92388713813338218541071789963, 3.40392273609469938900847431542, 3.99947599999390154890199283145, 4.69043599042521821225027955883, 4.80012424507694010567812018706, 5.49027878289581855034176344125, 5.72686912879348589735299017903, 6.35017905904169281722660168873, 6.35340026426278654169893536516, 7.16589876695208150011388504491, 7.42297558383710381328696778612, 8.095190877714211151420413806313, 8.188690276151103822548954148946, 8.727062162832495033338205550249, 9.134813727695630941301129936501
