L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s − 2·11-s + 12-s + 2·13-s + 15-s − 16-s + 4·17-s − 18-s + 20-s + 2·22-s − 23-s − 3·24-s + 25-s − 2·26-s − 27-s − 30-s + 2·31-s − 5·32-s + 2·33-s − 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 0.426·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.182·30-s + 0.359·31-s − 0.883·32-s + 0.348·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.82345045797645, −12.43869732952289, −12.10955408671096, −11.38475056397459, −11.03701637894903, −10.67321691855149, −10.11175607878924, −9.796840251412584, −9.332599860748971, −8.763974072321934, −8.166857914880482, −8.013883959005970, −7.492386699617595, −6.986149394931985, −6.345702554936307, −5.918308530939519, −5.186690022338921, −4.965974600041883, −4.404480621942580, −3.730003617249759, −3.385608995375661, −2.645437079196791, −1.740971082167407, −1.296901293840709, −0.6043207631599170, 0,
0.6043207631599170, 1.296901293840709, 1.740971082167407, 2.645437079196791, 3.385608995375661, 3.730003617249759, 4.404480621942580, 4.965974600041883, 5.186690022338921, 5.918308530939519, 6.345702554936307, 6.986149394931985, 7.492386699617595, 8.013883959005970, 8.166857914880482, 8.763974072321934, 9.332599860748971, 9.796840251412584, 10.11175607878924, 10.67321691855149, 11.03701637894903, 11.38475056397459, 12.10955408671096, 12.43869732952289, 12.82345045797645