| L(s) = 1 | + 2.69·2-s + 5.27·4-s + 4.13·7-s + 8.82·8-s + 11-s − 2.73·13-s + 11.1·14-s + 13.2·16-s + 2.41·17-s − 1.15·19-s + 2.69·22-s − 8.54·23-s − 7.38·26-s + 21.7·28-s + 1.67·29-s − 10.2·31-s + 18.1·32-s + 6.51·34-s − 5.71·37-s − 3.11·38-s + 9.11·41-s − 8.13·43-s + 5.27·44-s − 23.0·46-s + 1.47·47-s + 10.0·49-s − 14.4·52-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.63·4-s + 1.56·7-s + 3.12·8-s + 0.301·11-s − 0.759·13-s + 2.97·14-s + 3.31·16-s + 0.585·17-s − 0.264·19-s + 0.574·22-s − 1.78·23-s − 1.44·26-s + 4.11·28-s + 0.311·29-s − 1.84·31-s + 3.20·32-s + 1.11·34-s − 0.939·37-s − 0.504·38-s + 1.42·41-s − 1.24·43-s + 0.795·44-s − 3.39·46-s + 0.215·47-s + 1.43·49-s − 2.00·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.226618963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.226618963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 + 1.15T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.71T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 + 8.13T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 2.54T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 9.28T + 83T^{2} \) |
| 89 | \( 1 + 4.78T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.706636909231404378305941800268, −7.69185732812286323830096075988, −7.38756369457672930954538628989, −6.28824998874188112499132321851, −5.54718447721444237233852493365, −4.94458791896067904442947752482, −4.23804948933797513905201853873, −3.50920293945520800535221044206, −2.25564847160451577162259539000, −1.66212756500853651764085484082,
1.66212756500853651764085484082, 2.25564847160451577162259539000, 3.50920293945520800535221044206, 4.23804948933797513905201853873, 4.94458791896067904442947752482, 5.54718447721444237233852493365, 6.28824998874188112499132321851, 7.38756369457672930954538628989, 7.69185732812286323830096075988, 8.706636909231404378305941800268
