L(s) = 1 | − 2-s + 4-s − 5-s − 5.12·7-s − 8-s + 10-s − 5.12·11-s + 2·13-s + 5.12·14-s + 16-s − 7.12·17-s + 4·19-s − 20-s + 5.12·22-s − 23-s + 25-s − 2·26-s − 5.12·28-s − 2·29-s − 32-s + 7.12·34-s + 5.12·35-s − 7.12·37-s − 4·38-s + 40-s − 2·41-s − 5.12·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.93·7-s − 0.353·8-s + 0.316·10-s − 1.54·11-s + 0.554·13-s + 1.36·14-s + 0.250·16-s − 1.72·17-s + 0.917·19-s − 0.223·20-s + 1.09·22-s − 0.208·23-s + 0.200·25-s − 0.392·26-s − 0.968·28-s − 0.371·29-s − 0.176·32-s + 1.22·34-s + 0.865·35-s − 1.17·37-s − 0.648·38-s + 0.158·40-s − 0.312·41-s − 0.772·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4022858320\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4022858320\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 3.12T + 89T^{2} \) |
| 97 | \( 1 - 0.246T + 97T^{2} \) |
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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
−9.001689272658231811437568414186, −8.577001114223972297420009252660, −7.45048294540669792161821743024, −6.96847314861357414786976546877, −6.14012706877911246905249672678, −5.31607040833313925434530722999, −3.96803155340569292580337095906, −3.11439405559670064525366173300, −2.32612613328561852707480352135, −0.43167941363320819501824277025,
0.43167941363320819501824277025, 2.32612613328561852707480352135, 3.11439405559670064525366173300, 3.96803155340569292580337095906, 5.31607040833313925434530722999, 6.14012706877911246905249672678, 6.96847314861357414786976546877, 7.45048294540669792161821743024, 8.577001114223972297420009252660, 9.001689272658231811437568414186