| L(s) = 1 | − 2-s + 0.529·3-s + 4-s + 5-s − 0.529·6-s − 7-s − 8-s − 2.71·9-s − 10-s + 0.529·12-s + 4.81·13-s + 14-s + 0.529·15-s + 16-s − 6.02·17-s + 2.71·18-s − 3.59·19-s + 20-s − 0.529·21-s − 7.06·23-s − 0.529·24-s + 25-s − 4.81·26-s − 3.02·27-s − 28-s − 0.353·29-s − 0.529·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.305·3-s + 0.5·4-s + 0.447·5-s − 0.216·6-s − 0.377·7-s − 0.353·8-s − 0.906·9-s − 0.316·10-s + 0.152·12-s + 1.33·13-s + 0.267·14-s + 0.136·15-s + 0.250·16-s − 1.46·17-s + 0.641·18-s − 0.824·19-s + 0.223·20-s − 0.115·21-s − 1.47·23-s − 0.108·24-s + 0.200·25-s − 0.943·26-s − 0.582·27-s − 0.188·28-s − 0.0656·29-s − 0.0966·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108095798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108095798\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.529T + 3T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 6.02T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 + 7.06T + 23T^{2} \) |
| 29 | \( 1 + 0.353T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 + 9.35T + 41T^{2} \) |
| 43 | \( 1 + 9.13T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.64T + 53T^{2} \) |
| 59 | \( 1 - 2.78T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 + 8.04T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 1.23T + 97T^{2} \) |
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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
−8.145067399759047158875362417424, −7.02311330424738804953111563470, −6.41880582469074617588967206545, −5.99555074940147339503168688882, −5.18249164976202050948969985340, −3.99426194356017041608754436900, −3.46333657825402104351020432661, −2.31718787796917458322040865931, −1.95686254159834108161214656840, −0.53583477324452634195610478348,
0.53583477324452634195610478348, 1.95686254159834108161214656840, 2.31718787796917458322040865931, 3.46333657825402104351020432661, 3.99426194356017041608754436900, 5.18249164976202050948969985340, 5.99555074940147339503168688882, 6.41880582469074617588967206545, 7.02311330424738804953111563470, 8.145067399759047158875362417424
