| L(s) = 1 | − 2-s + 3·3-s − 7·4-s + 7·5-s − 3·6-s − 10·7-s + 15·8-s + 9·9-s − 7·10-s − 22·11-s − 21·12-s + 10·14-s + 21·15-s + 41·16-s + 37·17-s − 9·18-s + 30·19-s − 49·20-s − 30·21-s + 22·22-s − 162·23-s + 45·24-s − 76·25-s + 27·27-s + 70·28-s − 113·29-s − 21·30-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.626·5-s − 0.204·6-s − 0.539·7-s + 0.662·8-s + 1/3·9-s − 0.221·10-s − 0.603·11-s − 0.505·12-s + 0.190·14-s + 0.361·15-s + 0.640·16-s + 0.527·17-s − 0.117·18-s + 0.362·19-s − 0.547·20-s − 0.311·21-s + 0.213·22-s − 1.46·23-s + 0.382·24-s − 0.607·25-s + 0.192·27-s + 0.472·28-s − 0.723·29-s − 0.127·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 37 T + p^{3} T^{2} \) |
| 19 | \( 1 - 30 T + p^{3} T^{2} \) |
| 23 | \( 1 + 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 113 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 - 13 T + p^{3} T^{2} \) |
| 41 | \( 1 - 285 T + p^{3} T^{2} \) |
| 43 | \( 1 + 246 T + p^{3} T^{2} \) |
| 47 | \( 1 + 462 T + p^{3} T^{2} \) |
| 53 | \( 1 + 537 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 + 635 T + p^{3} T^{2} \) |
| 67 | \( 1 - 202 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 73 | \( 1 + 805 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 83 | \( 1 - 518 T + p^{3} T^{2} \) |
| 89 | \( 1 - 194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1202 T + p^{3} T^{2} \) |
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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
−9.763981882686809783399851470303, −9.436275112797533902347390413592, −8.216694699354464598045880342406, −7.72826019987774783598138872655, −6.30326089755494780999223939117, −5.32883037841768507513710932799, −4.16701869404045162900345307031, −3.03888161998064451376328152824, −1.62850739972349756762561155014, 0,
1.62850739972349756762561155014, 3.03888161998064451376328152824, 4.16701869404045162900345307031, 5.32883037841768507513710932799, 6.30326089755494780999223939117, 7.72826019987774783598138872655, 8.216694699354464598045880342406, 9.436275112797533902347390413592, 9.763981882686809783399851470303
