| L(s) = 1 | + 2-s + 4-s + 0.267·5-s + 8-s + 0.267·10-s − 6.19·11-s + 6.46·13-s + 16-s + 7·17-s − 0.732·19-s + 0.267·20-s − 6.19·22-s + 4.19·23-s − 4.92·25-s + 6.46·26-s − 1.53·29-s − 8.19·31-s + 32-s + 7·34-s + 10.6·37-s − 0.732·38-s + 0.267·40-s + 2.53·41-s − 1.46·43-s − 6.19·44-s + 4.19·46-s + 4.73·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.119·5-s + 0.353·8-s + 0.0847·10-s − 1.86·11-s + 1.79·13-s + 0.250·16-s + 1.69·17-s − 0.167·19-s + 0.0599·20-s − 1.32·22-s + 0.874·23-s − 0.985·25-s + 1.26·26-s − 0.285·29-s − 1.47·31-s + 0.176·32-s + 1.20·34-s + 1.75·37-s − 0.118·38-s + 0.0423·40-s + 0.396·41-s − 0.223·43-s − 0.934·44-s + 0.618·46-s + 0.690·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.493765844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.493765844\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 0.732T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 - 4.19T + 59T^{2} \) |
| 61 | \( 1 + 3.92T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 + 6.53T + 71T^{2} \) |
| 73 | \( 1 + 8.26T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.63154737740895890823531401902, −7.35062694648888552997509807419, −6.05193986313449556641568717912, −5.77623355945256934844047712964, −5.21927104765924035335365372707, −4.24528162004751063292812752860, −3.47672204224157539676236419226, −2.88682126596248651545546716194, −1.91446519504405065760906291454, −0.846087250399229237515631728529,
0.846087250399229237515631728529, 1.91446519504405065760906291454, 2.88682126596248651545546716194, 3.47672204224157539676236419226, 4.24528162004751063292812752860, 5.21927104765924035335365372707, 5.77623355945256934844047712964, 6.05193986313449556641568717912, 7.35062694648888552997509807419, 7.63154737740895890823531401902
