| L(s) = 1 | + 2.82·2-s + 2.39·3-s + 5.96·4-s + 6.74·6-s − 0.875·7-s + 11.1·8-s + 2.71·9-s − 3.86·11-s + 14.2·12-s + 0.582·13-s − 2.47·14-s + 19.6·16-s + 2.93·17-s + 7.67·18-s + 2.08·19-s − 2.09·21-s − 10.8·22-s − 0.349·23-s + 26.7·24-s + 1.64·26-s − 0.673·27-s − 5.22·28-s + 3.11·29-s − 4.04·31-s + 33.1·32-s − 9.23·33-s + 8.29·34-s + ⋯ |
| L(s) = 1 | + 1.99·2-s + 1.38·3-s + 2.98·4-s + 2.75·6-s − 0.330·7-s + 3.95·8-s + 0.906·9-s − 1.16·11-s + 4.11·12-s + 0.161·13-s − 0.660·14-s + 4.91·16-s + 0.712·17-s + 1.80·18-s + 0.478·19-s − 0.456·21-s − 2.32·22-s − 0.0728·23-s + 5.46·24-s + 0.322·26-s − 0.129·27-s − 0.987·28-s + 0.578·29-s − 0.726·31-s + 5.85·32-s − 1.60·33-s + 1.42·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.49006490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.49006490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.82T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 7 | \( 1 + 0.875T + 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 - 0.582T + 13T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 23 | \( 1 + 0.349T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 - 3.14T + 67T^{2} \) |
| 71 | \( 1 - 5.72T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 7.49T + 89T^{2} \) |
| 97 | \( 1 + 3.87T + 97T^{2} \) |
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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
−7.87688691913620510054473488101, −7.32328342663569943727793716278, −6.53841088730874613302410710933, −5.72764723058824474344117859481, −5.10433787487738314105296501986, −4.33807655240854818862797576619, −3.40901372773134727563101420522, −3.11469262723481546321191865711, −2.41316828758976807124709052798, −1.54603021724753158530807023558,
1.54603021724753158530807023558, 2.41316828758976807124709052798, 3.11469262723481546321191865711, 3.40901372773134727563101420522, 4.33807655240854818862797576619, 5.10433787487738314105296501986, 5.72764723058824474344117859481, 6.53841088730874613302410710933, 7.32328342663569943727793716278, 7.87688691913620510054473488101
