| L(s) = 1 | + 2.53·2-s + 4.41·4-s + 0.120·5-s + 0.879·7-s + 6.10·8-s + 0.305·10-s + 2.71·11-s − 5.10·13-s + 2.22·14-s + 6.63·16-s + 4.46·17-s + 2.87·19-s + 0.532·20-s + 6.87·22-s + 5.22·23-s − 4.98·25-s − 12.9·26-s + 3.87·28-s − 5.94·29-s + 1.42·31-s + 4.59·32-s + 11.3·34-s + 0.106·35-s − 10.5·37-s + 7.29·38-s + 0.736·40-s + 5.38·41-s + ⋯ |
| L(s) = 1 | + 1.79·2-s + 2.20·4-s + 0.0539·5-s + 0.332·7-s + 2.15·8-s + 0.0965·10-s + 0.819·11-s − 1.41·13-s + 0.595·14-s + 1.65·16-s + 1.08·17-s + 0.660·19-s + 0.118·20-s + 1.46·22-s + 1.08·23-s − 0.997·25-s − 2.53·26-s + 0.733·28-s − 1.10·29-s + 0.256·31-s + 0.812·32-s + 1.94·34-s + 0.0179·35-s − 1.73·37-s + 1.18·38-s + 0.116·40-s + 0.841·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.997220558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.997220558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 5 | \( 1 - 0.120T + 5T^{2} \) |
| 7 | \( 1 - 0.879T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 - 2.87T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 5.94T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 5.38T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 2.71T + 59T^{2} \) |
| 61 | \( 1 + 1.98T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 9.49T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 7.72T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 2.64T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−9.939173337794927270344994622168, −9.125983963949881766048969892689, −7.58516136863581889998845578290, −7.25465095390858026481375133991, −6.12403464313831684740738313643, −5.37355734183641939335254487774, −4.69008902325513668457534153049, −3.72097265799901943965579481945, −2.86025707375385917019174980777, −1.66216969591739152246595353089,
1.66216969591739152246595353089, 2.86025707375385917019174980777, 3.72097265799901943965579481945, 4.69008902325513668457534153049, 5.37355734183641939335254487774, 6.12403464313831684740738313643, 7.25465095390858026481375133991, 7.58516136863581889998845578290, 9.125983963949881766048969892689, 9.939173337794927270344994622168
