| L(s) = 1 | − 5-s − 1.52·7-s + 1.52·11-s + 0.622·13-s − 5.52·17-s + 2.90·19-s + 2.90·23-s + 25-s − 29-s − 2.90·31-s + 1.52·35-s + 2.47·37-s + 8.23·41-s − 5.65·43-s + 4.70·47-s − 4.67·49-s + 11.2·53-s − 1.52·55-s − 10.2·59-s − 7.93·61-s − 0.622·65-s − 2.90·67-s − 7.47·71-s + 2.90·73-s − 2.32·77-s − 16.5·79-s + 6.76·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.576·7-s + 0.459·11-s + 0.172·13-s − 1.34·17-s + 0.666·19-s + 0.605·23-s + 0.200·25-s − 0.185·29-s − 0.521·31-s + 0.257·35-s + 0.406·37-s + 1.28·41-s − 0.862·43-s + 0.686·47-s − 0.667·49-s + 1.55·53-s − 0.205·55-s − 1.33·59-s − 1.01·61-s − 0.0771·65-s − 0.354·67-s − 0.887·71-s + 0.339·73-s − 0.265·77-s − 1.86·79-s + 0.743·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 31 | \( 1 + 2.90T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 7.93T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−7.70387707206528897272644486440, −7.16932894359080537194244805663, −6.44598572432251328885130077396, −5.79171039778003899827925724927, −4.78893190492671786118497780606, −4.11366900073951580869716766929, −3.31821985756849302475115656690, −2.48965825817061695800959585562, −1.27447959105195894810472466583, 0,
1.27447959105195894810472466583, 2.48965825817061695800959585562, 3.31821985756849302475115656690, 4.11366900073951580869716766929, 4.78893190492671786118497780606, 5.79171039778003899827925724927, 6.44598572432251328885130077396, 7.16932894359080537194244805663, 7.70387707206528897272644486440
