| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 2·7-s − 8-s + 9-s − 10-s − 11-s + 12-s − 2·14-s + 15-s + 16-s − 18-s + 20-s + 2·21-s + 22-s + 4·23-s − 24-s + 25-s + 27-s + 2·28-s − 2·29-s − 30-s − 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s + 0.436·21-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.371·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.28871140460968, −13.44353972084008, −13.12134230313038, −12.70592834946349, −12.00682357149163, −11.38658734713367, −11.09444113667857, −10.56524068986207, −9.915062894618959, −9.632433113012775, −9.064452433270756, −8.430263313133153, −8.264428959194023, −7.619642816036750, −6.968408509941284, −6.783232937436458, −5.799655458701693, −5.426023463844463, −4.794531997210196, −4.153578275017175, −3.402475363478120, −2.820734725338237, −2.194971913658804, −1.620437871520514, −1.053006552819734, 0,
1.053006552819734, 1.620437871520514, 2.194971913658804, 2.820734725338237, 3.402475363478120, 4.153578275017175, 4.794531997210196, 5.426023463844463, 5.799655458701693, 6.783232937436458, 6.968408509941284, 7.619642816036750, 8.264428959194023, 8.430263313133153, 9.064452433270756, 9.632433113012775, 9.915062894618959, 10.56524068986207, 11.09444113667857, 11.38658734713367, 12.00682357149163, 12.70592834946349, 13.12134230313038, 13.44353972084008, 14.28871140460968
