| L(s) = 1 | + 0.863·2-s − 7.07·3-s − 31.2·4-s − 6.10·6-s + 36.7·7-s − 54.6·8-s − 192.·9-s − 302.·11-s + 221.·12-s + 373.·13-s + 31.7·14-s + 952.·16-s − 280.·17-s − 166.·18-s + 1.37e3·19-s − 259.·21-s − 261.·22-s − 1.86e3·23-s + 386.·24-s + 322.·26-s + 3.08e3·27-s − 1.14e3·28-s − 841·29-s + 1.47e3·31-s + 2.57e3·32-s + 2.13e3·33-s − 242.·34-s + ⋯ |
| L(s) = 1 | + 0.152·2-s − 0.453·3-s − 0.976·4-s − 0.0692·6-s + 0.283·7-s − 0.301·8-s − 0.794·9-s − 0.753·11-s + 0.443·12-s + 0.612·13-s + 0.0432·14-s + 0.930·16-s − 0.235·17-s − 0.121·18-s + 0.871·19-s − 0.128·21-s − 0.114·22-s − 0.733·23-s + 0.136·24-s + 0.0935·26-s + 0.814·27-s − 0.276·28-s − 0.185·29-s + 0.275·31-s + 0.443·32-s + 0.341·33-s − 0.0359·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 29 | \( 1 + 841T \) |
| good | 2 | \( 1 - 0.863T + 32T^{2} \) |
| 3 | \( 1 + 7.07T + 243T^{2} \) |
| 7 | \( 1 - 36.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 302.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 373.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 280.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.86e3T + 6.43e6T^{2} \) |
| 31 | \( 1 - 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.17e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.17e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.67e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.48e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.77e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.56e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.49e4T + 8.58e9T^{2} \) |
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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
−9.200267242495580417692306799275, −8.327853404842560579510574755827, −7.70742595860679503196890028732, −6.24547421599642155692489108342, −5.54038879430530207964787735326, −4.79350561065863311414686151428, −3.74692398104611569731743467722, −2.63249023762075256910718772819, −1.00484394244342486672503472313, 0,
1.00484394244342486672503472313, 2.63249023762075256910718772819, 3.74692398104611569731743467722, 4.79350561065863311414686151428, 5.54038879430530207964787735326, 6.24547421599642155692489108342, 7.70742595860679503196890028732, 8.327853404842560579510574755827, 9.200267242495580417692306799275
