| L(s) = 1 | + 2-s + 4-s − 2.39·5-s + 2.84·7-s + 8-s − 2.39·10-s + 5.71·11-s − 5.33·13-s + 2.84·14-s + 16-s − 5.68·17-s − 1.71·19-s − 2.39·20-s + 5.71·22-s + 6.10·23-s + 0.752·25-s − 5.33·26-s + 2.84·28-s − 2.56·29-s + 4.56·31-s + 32-s − 5.68·34-s − 6.82·35-s + 8.20·37-s − 1.71·38-s − 2.39·40-s + 0.830·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.07·5-s + 1.07·7-s + 0.353·8-s − 0.758·10-s + 1.72·11-s − 1.47·13-s + 0.760·14-s + 0.250·16-s − 1.37·17-s − 0.394·19-s − 0.536·20-s + 1.21·22-s + 1.27·23-s + 0.150·25-s − 1.04·26-s + 0.538·28-s − 0.475·29-s + 0.820·31-s + 0.176·32-s − 0.974·34-s − 1.15·35-s + 1.34·37-s − 0.278·38-s − 0.379·40-s + 0.129·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.071889423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.071889423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 - 5.71T + 11T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 1.71T + 19T^{2} \) |
| 23 | \( 1 - 6.10T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 4.56T + 31T^{2} \) |
| 37 | \( 1 - 8.20T + 37T^{2} \) |
| 41 | \( 1 - 0.830T + 41T^{2} \) |
| 43 | \( 1 - 3.47T + 43T^{2} \) |
| 47 | \( 1 + 3.29T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 + 4.62T + 89T^{2} \) |
| 97 | \( 1 - 8.89T + 97T^{2} \) |
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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
−7.73391693940000281885148776996, −6.97187922748620705946678226720, −6.68785069158154406507707201313, −5.61214889779382712701761515679, −4.67608802812067468714634033142, −4.40576575971214188224296859011, −3.80986731104163665357720286052, −2.71088674381497648354291684450, −1.92830212273327225860354790078, −0.78734519350656327787581611336,
0.78734519350656327787581611336, 1.92830212273327225860354790078, 2.71088674381497648354291684450, 3.80986731104163665357720286052, 4.40576575971214188224296859011, 4.67608802812067468714634033142, 5.61214889779382712701761515679, 6.68785069158154406507707201313, 6.97187922748620705946678226720, 7.73391693940000281885148776996
