L(s) = 1 | − 2.46·2-s + 3.43·3-s + 4.09·4-s − 5-s − 8.48·6-s − 7-s − 5.17·8-s + 8.80·9-s + 2.46·10-s + 3.46·11-s + 14.0·12-s − 6.33·13-s + 2.46·14-s − 3.43·15-s + 4.59·16-s + 4.14·17-s − 21.7·18-s + 0.869·19-s − 4.09·20-s − 3.43·21-s − 8.56·22-s + 23-s − 17.7·24-s + 25-s + 15.6·26-s + 19.9·27-s − 4.09·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 1.98·3-s + 2.04·4-s − 0.447·5-s − 3.46·6-s − 0.377·7-s − 1.83·8-s + 2.93·9-s + 0.780·10-s + 1.04·11-s + 4.06·12-s − 1.75·13-s + 0.659·14-s − 0.887·15-s + 1.14·16-s + 1.00·17-s − 5.12·18-s + 0.199·19-s − 0.916·20-s − 0.749·21-s − 1.82·22-s + 0.208·23-s − 3.63·24-s + 0.200·25-s + 3.06·26-s + 3.84·27-s − 0.774·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.337044376\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.337044376\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 3.43T + 3T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 6.33T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 0.869T + 19T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 + 0.497T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 - 3.66T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−9.766744305217384017004694025187, −9.450415495199355249991147178423, −8.566223141023952458372588717941, −7.987675265041535396935360896815, −7.25631154510710396658800119698, −6.73294296213277415250833242254, −4.51032730725226319879042202507, −3.23978691396394718206004432314, −2.46958088612263851227764791213, −1.22284036712109489824507076188,
1.22284036712109489824507076188, 2.46958088612263851227764791213, 3.23978691396394718206004432314, 4.51032730725226319879042202507, 6.73294296213277415250833242254, 7.25631154510710396658800119698, 7.987675265041535396935360896815, 8.566223141023952458372588717941, 9.450415495199355249991147178423, 9.766744305217384017004694025187