L(s) = 1 | + 2-s + 4-s − 2.45·5-s − 3.97·7-s + 8-s − 2.45·10-s + 3.25·11-s + 2.63·13-s − 3.97·14-s + 16-s + 7.31·17-s − 5.34·19-s − 2.45·20-s + 3.25·22-s + 6.94·23-s + 1.05·25-s + 2.63·26-s − 3.97·28-s − 7.73·29-s − 3.43·31-s + 32-s + 7.31·34-s + 9.77·35-s − 5.84·37-s − 5.34·38-s − 2.45·40-s − 9.31·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.10·5-s − 1.50·7-s + 0.353·8-s − 0.777·10-s + 0.981·11-s + 0.730·13-s − 1.06·14-s + 0.250·16-s + 1.77·17-s − 1.22·19-s − 0.550·20-s + 0.694·22-s + 1.44·23-s + 0.210·25-s + 0.516·26-s − 0.751·28-s − 1.43·29-s − 0.616·31-s + 0.176·32-s + 1.25·34-s + 1.65·35-s − 0.961·37-s − 0.866·38-s − 0.388·40-s − 1.45·41-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
\(L(1)\) |
\(\approx\) |
\(2.073032688\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.073032688\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.45T + 5T^{2} \) |
| 7 | \( 1 + 3.97T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 - 7.31T + 17T^{2} \) |
| 19 | \( 1 + 5.34T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 + 9.31T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 6.59T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 - 4.36T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.59572376377654543400570050890, −7.01314106731230060324193864734, −6.48179594623270497245571146470, −5.79136324261308335697862922660, −5.03160528538708761125310707344, −3.92835684147812504543221192903, −3.53833514402871705798648149448, −3.23893964943142900813577950402, −1.83671023490885035928330871462, −0.63490290592735311254655564490,
0.63490290592735311254655564490, 1.83671023490885035928330871462, 3.23893964943142900813577950402, 3.53833514402871705798648149448, 3.92835684147812504543221192903, 5.03160528538708761125310707344, 5.79136324261308335697862922660, 6.48179594623270497245571146470, 7.01314106731230060324193864734, 7.59572376377654543400570050890