| L(s) = 1 | − 1.48·2-s + 0.193·4-s − 2.48·7-s + 2.67·8-s − 2·11-s + 5.83·13-s + 3.67·14-s − 4.35·16-s + 3.76·17-s + 2.96·22-s + 0.806·23-s − 8.63·26-s − 0.481·28-s − 4.06·29-s + 31-s + 1.09·32-s − 5.58·34-s − 12.1·37-s − 9.92·41-s + 10.3·43-s − 0.387·44-s − 1.19·46-s − 1.58·47-s − 0.843·49-s + 1.13·52-s − 7.11·53-s − 6.63·56-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0969·4-s − 0.937·7-s + 0.945·8-s − 0.603·11-s + 1.61·13-s + 0.982·14-s − 1.08·16-s + 0.913·17-s + 0.631·22-s + 0.168·23-s − 1.69·26-s − 0.0909·28-s − 0.754·29-s + 0.179·31-s + 0.193·32-s − 0.957·34-s − 1.99·37-s − 1.54·41-s + 1.57·43-s − 0.0584·44-s − 0.176·46-s − 0.230·47-s − 0.120·49-s + 0.156·52-s − 0.977·53-s − 0.886·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 0.806T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 + 7.11T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 1.25T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 0.0303T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−7.83085096376450162095131925784, −7.03384945539502868544046238666, −6.35963975527518870178408288859, −5.58442063299519439737090647975, −4.82819335139974406541011627274, −3.66977219552438767209883282014, −3.31015447868465147053041711440, −1.97258035879966970335370864856, −1.06499528321804395605277008259, 0,
1.06499528321804395605277008259, 1.97258035879966970335370864856, 3.31015447868465147053041711440, 3.66977219552438767209883282014, 4.82819335139974406541011627274, 5.58442063299519439737090647975, 6.35963975527518870178408288859, 7.03384945539502868544046238666, 7.83085096376450162095131925784
