L(s) = 1 | + 1.21·2-s + 2.36·3-s − 0.515·4-s + 2.87·6-s + 0.846·7-s − 3.06·8-s + 2.58·9-s − 11-s − 1.21·12-s + 0.935·13-s + 1.03·14-s − 2.70·16-s + 17-s + 3.14·18-s + 5.75·19-s + 2·21-s − 1.21·22-s − 2.54·23-s − 7.24·24-s + 1.13·26-s − 0.990·27-s − 0.436·28-s + 9.45·29-s + 4.12·31-s + 2.83·32-s − 2.36·33-s + 1.21·34-s + ⋯ |
L(s) = 1 | + 0.861·2-s + 1.36·3-s − 0.257·4-s + 1.17·6-s + 0.319·7-s − 1.08·8-s + 0.860·9-s − 0.301·11-s − 0.351·12-s + 0.259·13-s + 0.275·14-s − 0.675·16-s + 0.242·17-s + 0.741·18-s + 1.32·19-s + 0.436·21-s − 0.259·22-s − 0.531·23-s − 1.47·24-s + 0.223·26-s − 0.190·27-s − 0.0825·28-s + 1.75·29-s + 0.741·31-s + 0.501·32-s − 0.411·33-s + 0.208·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.501001345\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.501001345\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 3 | \( 1 - 2.36T + 3T^{2} \) |
| 7 | \( 1 - 0.846T + 7T^{2} \) |
| 13 | \( 1 - 0.935T + 13T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 - 9.45T + 29T^{2} \) |
| 31 | \( 1 - 4.12T + 31T^{2} \) |
| 37 | \( 1 - 0.776T + 37T^{2} \) |
| 41 | \( 1 + 2.54T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 9.72T + 61T^{2} \) |
| 67 | \( 1 + 1.13T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 - 9.13T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 9.10T + 83T^{2} \) |
| 89 | \( 1 + 0.835T + 89T^{2} \) |
| 97 | \( 1 + 8.21T + 97T^{2} \) |
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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
−8.227305018659894591812886978055, −7.84534792311009503447869099393, −6.87996653860080187659039300898, −5.93613894518723919906825109512, −5.22683762668709111034486538608, −4.41824722826831634403782856247, −3.73856900327122958065009695992, −2.97534975684993375382473718888, −2.42345514140626795396446306676, −1.00834730735439999590031028305,
1.00834730735439999590031028305, 2.42345514140626795396446306676, 2.97534975684993375382473718888, 3.73856900327122958065009695992, 4.41824722826831634403782856247, 5.22683762668709111034486538608, 5.93613894518723919906825109512, 6.87996653860080187659039300898, 7.84534792311009503447869099393, 8.227305018659894591812886978055