| L(s) = 1 | + 2-s + 1.29·3-s + 4-s + 4.10·5-s + 1.29·6-s − 1.09·7-s + 8-s − 1.33·9-s + 4.10·10-s − 0.113·11-s + 1.29·12-s + 0.757·13-s − 1.09·14-s + 5.29·15-s + 16-s − 5.71·17-s − 1.33·18-s + 5.76·19-s + 4.10·20-s − 1.41·21-s − 0.113·22-s + 23-s + 1.29·24-s + 11.8·25-s + 0.757·26-s − 5.59·27-s − 1.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.744·3-s + 0.5·4-s + 1.83·5-s + 0.526·6-s − 0.415·7-s + 0.353·8-s − 0.445·9-s + 1.29·10-s − 0.0341·11-s + 0.372·12-s + 0.210·13-s − 0.293·14-s + 1.36·15-s + 0.250·16-s − 1.38·17-s − 0.314·18-s + 1.32·19-s + 0.918·20-s − 0.309·21-s − 0.0241·22-s + 0.208·23-s + 0.263·24-s + 2.37·25-s + 0.148·26-s − 1.07·27-s − 0.207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.147670571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147670571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 + 0.113T + 11T^{2} \) |
| 13 | \( 1 - 0.757T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + 9.56T + 43T^{2} \) |
| 47 | \( 1 - 6.64T + 47T^{2} \) |
| 53 | \( 1 + 8.15T + 53T^{2} \) |
| 59 | \( 1 + 9.27T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 5.56T + 83T^{2} \) |
| 89 | \( 1 - 6.56T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−9.527919324218085314049184271439, −9.022063659195326407023046558522, −8.085234470748652365493500978037, −6.83975521762219547922674354314, −6.24692681573739805959445370914, −5.49205999554373918308653410721, −4.62510993596660084356545057241, −3.17330835425174528622979698540, −2.62247696977359216881977682237, −1.60160779234231749473945062082,
1.60160779234231749473945062082, 2.62247696977359216881977682237, 3.17330835425174528622979698540, 4.62510993596660084356545057241, 5.49205999554373918308653410721, 6.24692681573739805959445370914, 6.83975521762219547922674354314, 8.085234470748652365493500978037, 9.022063659195326407023046558522, 9.527919324218085314049184271439
