| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s + 12-s + 6·13-s + 4·14-s + 16-s − 2·17-s − 18-s + 19-s − 4·21-s − 8·23-s − 24-s − 6·26-s + 27-s − 4·28-s + 2·29-s − 8·31-s − 32-s + 2·34-s + 36-s − 10·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.872·21-s − 1.66·23-s − 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s − 1.64·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.585925603894540060565715104804, −7.76563036209357695024954698616, −6.94918005670784987807744749260, −6.25114320584730063899223438084, −5.68416987226466468396646530929, −4.03099478517209644382697426526, −3.53340927772324168849803703633, −2.58873352488295884524264659694, −1.47474897704742341732353852406, 0,
1.47474897704742341732353852406, 2.58873352488295884524264659694, 3.53340927772324168849803703633, 4.03099478517209644382697426526, 5.68416987226466468396646530929, 6.25114320584730063899223438084, 6.94918005670784987807744749260, 7.76563036209357695024954698616, 8.585925603894540060565715104804
