| L(s) = 1 | − 0.763·2-s + 0.889·3-s − 1.41·4-s − 5-s − 0.678·6-s + 2.40·7-s + 2.60·8-s − 2.20·9-s + 0.763·10-s − 11-s − 1.26·12-s − 5.68·13-s − 1.83·14-s − 0.889·15-s + 0.845·16-s − 2.70·17-s + 1.68·18-s + 7.04·19-s + 1.41·20-s + 2.14·21-s + 0.763·22-s + 2.82·23-s + 2.31·24-s + 25-s + 4.33·26-s − 4.63·27-s − 3.41·28-s + ⋯ |
| L(s) = 1 | − 0.539·2-s + 0.513·3-s − 0.708·4-s − 0.447·5-s − 0.277·6-s + 0.910·7-s + 0.922·8-s − 0.736·9-s + 0.241·10-s − 0.301·11-s − 0.363·12-s − 1.57·13-s − 0.491·14-s − 0.229·15-s + 0.211·16-s − 0.656·17-s + 0.397·18-s + 1.61·19-s + 0.317·20-s + 0.467·21-s + 0.162·22-s + 0.589·23-s + 0.473·24-s + 0.200·25-s + 0.851·26-s − 0.891·27-s − 0.645·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 0.763T + 2T^{2} \) |
| 3 | \( 1 - 0.889T + 3T^{2} \) |
| 7 | \( 1 - 2.40T + 7T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 - 7.04T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 - 8.59T + 31T^{2} \) |
| 37 | \( 1 + 5.66T + 37T^{2} \) |
| 41 | \( 1 + 6.06T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 4.40T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.172309009547133872331887208616, −7.60464280610458880191738312296, −7.02972362783632923019337308267, −5.64853003374889353174332652974, −4.80981283348010149689146620325, −4.56865313260474997407895691943, −3.23335010344901499843455675758, −2.53633515843208848684073525654, −1.24629734621620421051659624340, 0,
1.24629734621620421051659624340, 2.53633515843208848684073525654, 3.23335010344901499843455675758, 4.56865313260474997407895691943, 4.80981283348010149689146620325, 5.64853003374889353174332652974, 7.02972362783632923019337308267, 7.60464280610458880191738312296, 8.172309009547133872331887208616
