| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 4·8-s + 3·9-s − 2·10-s − 11-s + 2·12-s + 8·13-s − 15-s + 8·16-s + 2·17-s + 6·18-s − 2·20-s − 2·22-s + 23-s + 4·24-s + 5·25-s + 16·26-s + 8·27-s − 2·30-s − 7·31-s + 8·32-s − 33-s + 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1.41·8-s + 9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s + 2.21·13-s − 0.258·15-s + 2·16-s + 0.485·17-s + 1.41·18-s − 0.447·20-s − 0.426·22-s + 0.208·23-s + 0.816·24-s + 25-s + 3.13·26-s + 1.53·27-s − 0.365·30-s − 1.25·31-s + 1.41·32-s − 0.174·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.753186635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.753186635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.99960232406387512276992073601, −10.79442945066406867786427534713, −10.11208183022591151771880337247, −10.10161325232878934977793905390, −9.187806742001175353798185728071, −8.648243581903823590806133116743, −8.160990163992078280327318046859, −8.160739886601978036544926623006, −7.18859000729023107478828220508, −6.94202341810636088523073471072, −6.58650405740226903603653941299, −5.76401693069603270053043269476, −5.38491769674850279670869745872, −4.68830235543696078580726996581, −4.50798153946797896833286686474, −3.63787617343264988492771006092, −3.47857556716492841054601252798, −3.03734137867374490769487646022, −1.64381935873697940528143599926, −1.43200026555227946777253848653,
1.43200026555227946777253848653, 1.64381935873697940528143599926, 3.03734137867374490769487646022, 3.47857556716492841054601252798, 3.63787617343264988492771006092, 4.50798153946797896833286686474, 4.68830235543696078580726996581, 5.38491769674850279670869745872, 5.76401693069603270053043269476, 6.58650405740226903603653941299, 6.94202341810636088523073471072, 7.18859000729023107478828220508, 8.160739886601978036544926623006, 8.160990163992078280327318046859, 8.648243581903823590806133116743, 9.187806742001175353798185728071, 10.10161325232878934977793905390, 10.11208183022591151771880337247, 10.79442945066406867786427534713, 10.99960232406387512276992073601
