| L(s) = 1 | + 2-s − 3-s + 4-s − 2.05·5-s − 6-s + 8-s + 9-s − 2.05·10-s + 2.36·11-s − 12-s + 4.98·13-s + 2.05·15-s + 16-s + 0.0143·17-s + 18-s − 19-s − 2.05·20-s + 2.36·22-s + 7.34·23-s − 24-s − 0.778·25-s + 4.98·26-s − 27-s − 8.46·29-s + 2.05·30-s − 1.54·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.918·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.649·10-s + 0.714·11-s − 0.288·12-s + 1.38·13-s + 0.530·15-s + 0.250·16-s + 0.00347·17-s + 0.235·18-s − 0.229·19-s − 0.459·20-s + 0.505·22-s + 1.53·23-s − 0.204·24-s − 0.155·25-s + 0.976·26-s − 0.192·27-s − 1.57·29-s + 0.375·30-s − 0.277·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.377649199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.377649199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.05T + 5T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 17 | \( 1 - 0.0143T + 17T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 + 8.46T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 - 9.88T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 5.17T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 + 8.99T + 59T^{2} \) |
| 61 | \( 1 + 5.36T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 8.10T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.86430299208347907544066672670, −7.40034077976913244642880095112, −6.50718822540993689402126923502, −6.05912488741391904099487604282, −5.21785849302446238487678057884, −4.38514540459620185905133917899, −3.79182501988648576970481218259, −3.19197266347394780745939485577, −1.80689733939929103081540324152, −0.790437486101857919170642129070,
0.790437486101857919170642129070, 1.80689733939929103081540324152, 3.19197266347394780745939485577, 3.79182501988648576970481218259, 4.38514540459620185905133917899, 5.21785849302446238487678057884, 6.05912488741391904099487604282, 6.50718822540993689402126923502, 7.40034077976913244642880095112, 7.86430299208347907544066672670
