| L(s) = 1 | − 1.77·2-s − 3-s + 1.13·4-s + 1.77·6-s + 0.437·7-s + 1.52·8-s + 9-s + 5.73·11-s − 1.13·12-s − 13-s − 0.775·14-s − 4.98·16-s − 3.98·17-s − 1.77·18-s + 4.77·19-s − 0.437·21-s − 10.1·22-s − 0.337·23-s − 1.52·24-s + 1.77·26-s − 27-s + 0.498·28-s + 1.72·29-s − 7.86·31-s + 5.77·32-s − 5.73·33-s + 7.05·34-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.569·4-s + 0.723·6-s + 0.165·7-s + 0.539·8-s + 0.333·9-s + 1.72·11-s − 0.328·12-s − 0.277·13-s − 0.207·14-s − 1.24·16-s − 0.965·17-s − 0.417·18-s + 1.09·19-s − 0.0954·21-s − 2.16·22-s − 0.0704·23-s − 0.311·24-s + 0.347·26-s − 0.192·27-s + 0.0941·28-s + 0.319·29-s − 1.41·31-s + 1.02·32-s − 0.998·33-s + 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6927105709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6927105709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 7 | \( 1 - 0.437T + 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 + 0.337T + 23T^{2} \) |
| 29 | \( 1 - 1.72T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + 1.66T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 + 2.17T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 2.55T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 5.41T + 83T^{2} \) |
| 89 | \( 1 + 4.72T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−9.877295357321798460169870133968, −9.119000047089896822972058287346, −8.676370196693719652071617253423, −7.38687092899236568022638884956, −6.96843000152286584716324202159, −5.89644476743077741521678158499, −4.72482426881556293532939019927, −3.80144542670960684276795168159, −1.96094184077431287355880251811, −0.845479417679410831751563583556,
0.845479417679410831751563583556, 1.96094184077431287355880251811, 3.80144542670960684276795168159, 4.72482426881556293532939019927, 5.89644476743077741521678158499, 6.96843000152286584716324202159, 7.38687092899236568022638884956, 8.676370196693719652071617253423, 9.119000047089896822972058287346, 9.877295357321798460169870133968
