| L(s) = 1 | − 2-s + 2.20·3-s + 4-s − 2.34·5-s − 2.20·6-s − 4.48·7-s − 8-s + 1.85·9-s + 2.34·10-s − 11-s + 2.20·12-s − 4.37·13-s + 4.48·14-s − 5.16·15-s + 16-s + 4.01·17-s − 1.85·18-s + 3.28·19-s − 2.34·20-s − 9.88·21-s + 22-s − 7.01·23-s − 2.20·24-s + 0.494·25-s + 4.37·26-s − 2.52·27-s − 4.48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.27·3-s + 0.5·4-s − 1.04·5-s − 0.899·6-s − 1.69·7-s − 0.353·8-s + 0.617·9-s + 0.741·10-s − 0.301·11-s + 0.635·12-s − 1.21·13-s + 1.19·14-s − 1.33·15-s + 0.250·16-s + 0.974·17-s − 0.436·18-s + 0.754·19-s − 0.524·20-s − 2.15·21-s + 0.213·22-s − 1.46·23-s − 0.449·24-s + 0.0989·25-s + 0.858·26-s − 0.486·27-s − 0.848·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8250329947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8250329947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 13 | \( 1 + 4.37T + 13T^{2} \) |
| 17 | \( 1 - 4.01T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 7.01T + 23T^{2} \) |
| 29 | \( 1 - 0.170T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 - 8.69T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 + 6.34T + 47T^{2} \) |
| 53 | \( 1 - 1.69T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 9.02T + 61T^{2} \) |
| 67 | \( 1 + 1.77T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 - 5.59T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.190180076046641024967466955668, −7.71888297324148288643856910379, −7.41220654741289043612145503640, −6.41402825277577060359543728460, −5.59973848044120843522558402001, −4.24692855158444291690447381740, −3.45921455045729031720288002009, −2.97885434810193159003730190454, −2.19534483269528498100997546931, −0.49798133627890207503902958122,
0.49798133627890207503902958122, 2.19534483269528498100997546931, 2.97885434810193159003730190454, 3.45921455045729031720288002009, 4.24692855158444291690447381740, 5.59973848044120843522558402001, 6.41402825277577060359543728460, 7.41220654741289043612145503640, 7.71888297324148288643856910379, 8.190180076046641024967466955668
