| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 4·11-s − 12-s + 2·13-s + 2·15-s + 16-s − 17-s + 18-s + 4·19-s − 2·20-s + 4·22-s − 24-s − 25-s + 2·26-s − 27-s + 10·29-s + 2·30-s + 32-s − 4·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.85·29-s + 0.365·30-s + 0.176·32-s − 0.696·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.02765129211432, −13.46644679543782, −13.10208797855788, −12.31799006824815, −12.10036481369903, −11.54616458889656, −11.37380905146343, −10.83710843111859, −10.09767607532009, −9.729348403885083, −9.039495561833026, −8.400251634635901, −7.987460706508780, −7.355368250562008, −6.753310872403035, −6.460387305830343, −5.908345721431090, −5.224440951819917, −4.693902052352477, −4.138759265332021, −3.755962643443788, −3.147499703646503, −2.467542566714234, −1.387223626743176, −1.070254240076523, 0,
1.070254240076523, 1.387223626743176, 2.467542566714234, 3.147499703646503, 3.755962643443788, 4.138759265332021, 4.693902052352477, 5.224440951819917, 5.908345721431090, 6.460387305830343, 6.753310872403035, 7.355368250562008, 7.987460706508780, 8.400251634635901, 9.039495561833026, 9.729348403885083, 10.09767607532009, 10.83710843111859, 11.37380905146343, 11.54616458889656, 12.10036481369903, 12.31799006824815, 13.10208797855788, 13.46644679543782, 14.02765129211432
