| L(s) = 1 | + 2-s + 1.19·3-s + 4-s + 1.19·6-s + 7-s + 8-s − 1.56·9-s − 1.63·11-s + 1.19·12-s − 4.03·13-s + 14-s + 16-s − 17-s − 1.56·18-s − 0.530·19-s + 1.19·21-s − 1.63·22-s + 0.469·23-s + 1.19·24-s − 4.03·26-s − 5.46·27-s + 28-s − 8.50·29-s + 3.19·31-s + 32-s − 1.96·33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.692·3-s + 0.5·4-s + 0.489·6-s + 0.377·7-s + 0.353·8-s − 0.521·9-s − 0.493·11-s + 0.346·12-s − 1.11·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s − 0.368·18-s − 0.121·19-s + 0.261·21-s − 0.348·22-s + 0.0979·23-s + 0.244·24-s − 0.790·26-s − 1.05·27-s + 0.188·28-s − 1.57·29-s + 0.574·31-s + 0.176·32-s − 0.341·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 + 4.03T + 13T^{2} \) |
| 19 | \( 1 + 0.530T + 19T^{2} \) |
| 23 | \( 1 - 0.469T + 23T^{2} \) |
| 29 | \( 1 + 8.50T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 7.39T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 - 5.53T + 47T^{2} \) |
| 53 | \( 1 + 1.86T + 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 + 4.19T + 67T^{2} \) |
| 71 | \( 1 - 9.56T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 1.59T + 83T^{2} \) |
| 89 | \( 1 - 3.53T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−7.69104031458190757312854451740, −7.11713802984243416089732655351, −6.26862028060929083962082098935, −5.29260162986037594682810389737, −5.00915976442575707831023101359, −3.94814243882095579636734298179, −3.23442105756907132434121957046, −2.43797592567954518120177124756, −1.79576931602176565823106156600, 0,
1.79576931602176565823106156600, 2.43797592567954518120177124756, 3.23442105756907132434121957046, 3.94814243882095579636734298179, 5.00915976442575707831023101359, 5.29260162986037594682810389737, 6.26862028060929083962082098935, 7.11713802984243416089732655351, 7.69104031458190757312854451740
