| L(s) = 1 | + 4·2-s − 25.5·3-s + 16·4-s − 25·5-s − 102.·6-s − 185.·7-s + 64·8-s + 412.·9-s − 100·10-s − 190.·11-s − 409.·12-s − 654.·13-s − 742.·14-s + 639.·15-s + 256·16-s + 493.·17-s + 1.64e3·18-s + 3.10e3·19-s − 400·20-s + 4.75e3·21-s − 762.·22-s − 1.05e3·23-s − 1.63e3·24-s + 625·25-s − 2.61e3·26-s − 4.32e3·27-s − 2.97e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.64·3-s + 0.5·4-s − 0.447·5-s − 1.16·6-s − 1.43·7-s + 0.353·8-s + 1.69·9-s − 0.316·10-s − 0.475·11-s − 0.820·12-s − 1.07·13-s − 1.01·14-s + 0.734·15-s + 0.250·16-s + 0.414·17-s + 1.19·18-s + 1.97·19-s − 0.223·20-s + 2.35·21-s − 0.335·22-s − 0.415·23-s − 0.580·24-s + 0.200·25-s − 0.759·26-s − 1.14·27-s − 0.716·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 5 | \( 1 + 25T \) |
| 79 | \( 1 - 6.24e3T \) |
| good | 3 | \( 1 + 25.5T + 243T^{2} \) |
| 7 | \( 1 + 185.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 190.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 654.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 493.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 3.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 41.3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.93e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.66e4T + 2.07e9T^{2} \) |
| 83 | \( 1 - 1.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.62e5T + 8.58e9T^{2} \) |
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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
−9.658631888736555957320848137801, −7.80149249093272673843169388050, −7.04077581073235154238375742405, −6.33979868406750880158708057609, −5.46175737254929939102677925735, −4.90075141533921711743129224079, −3.70588239925721475976469724715, −2.73717557284280560093751284617, −0.921583216755717072872085077939, 0,
0.921583216755717072872085077939, 2.73717557284280560093751284617, 3.70588239925721475976469724715, 4.90075141533921711743129224079, 5.46175737254929939102677925735, 6.33979868406750880158708057609, 7.04077581073235154238375742405, 7.80149249093272673843169388050, 9.658631888736555957320848137801
