| L(s) = 1 | + 2.67·2-s + 5.13·4-s + 2.04·7-s + 8.37·8-s − 3.94·11-s + 5.35·13-s + 5.47·14-s + 12.1·16-s + 6.45·17-s − 19-s − 10.5·22-s + 3.48·23-s + 14.2·26-s + 10.5·28-s − 8.34·29-s − 2.96·31-s + 15.5·32-s + 17.2·34-s − 6.42·37-s − 2.67·38-s − 11.4·41-s + 7.22·43-s − 20.2·44-s + 9.30·46-s − 1.11·47-s − 2.79·49-s + 27.4·52-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.56·4-s + 0.774·7-s + 2.96·8-s − 1.18·11-s + 1.48·13-s + 1.46·14-s + 3.02·16-s + 1.56·17-s − 0.229·19-s − 2.24·22-s + 0.726·23-s + 2.80·26-s + 1.98·28-s − 1.55·29-s − 0.532·31-s + 2.75·32-s + 2.95·34-s − 1.05·37-s − 0.433·38-s − 1.78·41-s + 1.10·43-s − 3.05·44-s + 1.37·46-s − 0.162·47-s − 0.399·49-s + 3.81·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.763111532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.763111532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 6.45T + 17T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 8.34T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 + 6.42T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 + 0.362T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 + 0.135T + 61T^{2} \) |
| 67 | \( 1 - 5.17T + 67T^{2} \) |
| 71 | \( 1 + 2.87T + 71T^{2} \) |
| 73 | \( 1 + 1.87T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 7.20T + 83T^{2} \) |
| 89 | \( 1 - 6.41T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
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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
−7.975605684205865490278332296584, −7.56628085038168036862367676624, −6.68406595537968648635663540548, −5.81597936297339861443996882199, −5.32480289081274748678967093222, −4.83716895880217908174947432173, −3.62316568189625322638780271320, −3.41881642415596382139200200215, −2.21977329315223754556647793164, −1.37848502534543186934465601582,
1.37848502534543186934465601582, 2.21977329315223754556647793164, 3.41881642415596382139200200215, 3.62316568189625322638780271320, 4.83716895880217908174947432173, 5.32480289081274748678967093222, 5.81597936297339861443996882199, 6.68406595537968648635663540548, 7.56628085038168036862367676624, 7.975605684205865490278332296584
