| L(s) = 1 | − 2-s − 8·3-s − 7·4-s − 5·5-s + 8·6-s − 14·7-s + 15·8-s + 37·9-s + 5·10-s + 20·11-s + 56·12-s − 58·13-s + 14·14-s + 40·15-s + 41·16-s − 37·18-s − 80·19-s + 35·20-s + 112·21-s − 20·22-s + 118·23-s − 120·24-s + 25·25-s + 58·26-s − 80·27-s + 98·28-s − 126·29-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.53·3-s − 7/8·4-s − 0.447·5-s + 0.544·6-s − 0.755·7-s + 0.662·8-s + 1.37·9-s + 0.158·10-s + 0.548·11-s + 1.34·12-s − 1.23·13-s + 0.267·14-s + 0.688·15-s + 0.640·16-s − 0.484·18-s − 0.965·19-s + 0.391·20-s + 1.16·21-s − 0.193·22-s + 1.06·23-s − 1.02·24-s + 1/5·25-s + 0.437·26-s − 0.570·27-s + 0.661·28-s − 0.806·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 - 118 T + p^{3} T^{2} \) |
| 29 | \( 1 + 126 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 37 | \( 1 - 134 T + p^{3} T^{2} \) |
| 41 | \( 1 + 100 T + p^{3} T^{2} \) |
| 43 | \( 1 + 272 T + p^{3} T^{2} \) |
| 47 | \( 1 + 464 T + p^{3} T^{2} \) |
| 53 | \( 1 + 642 T + p^{3} T^{2} \) |
| 59 | \( 1 - 180 T + p^{3} T^{2} \) |
| 61 | \( 1 - 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 924 T + p^{3} T^{2} \) |
| 71 | \( 1 + 90 T + p^{3} T^{2} \) |
| 73 | \( 1 + 828 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1334 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1490 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1376 T + p^{3} T^{2} \) |
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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
−8.438780444835476103396925133742, −7.36605102979008847560407456923, −6.71712403084333860779528424672, −5.84758600797573662152055507883, −4.90077707298637885327358585168, −4.41336386404945379777534417769, −3.24046301547127520017105891300, −1.42523980128298221021349368133, 0, 0,
1.42523980128298221021349368133, 3.24046301547127520017105891300, 4.41336386404945379777534417769, 4.90077707298637885327358585168, 5.84758600797573662152055507883, 6.71712403084333860779528424672, 7.36605102979008847560407456923, 8.438780444835476103396925133742
