| L(s) = 1 | − 3·2-s + 4-s − 5·5-s + 7·7-s + 21·8-s + 15·10-s − 60·11-s + 38·13-s − 21·14-s − 71·16-s + 84·17-s + 110·19-s − 5·20-s + 180·22-s − 120·23-s + 25·25-s − 114·26-s + 7·28-s − 162·29-s + 236·31-s + 45·32-s − 252·34-s − 35·35-s − 376·37-s − 330·38-s − 105·40-s + 126·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s − 0.447·5-s + 0.377·7-s + 0.928·8-s + 0.474·10-s − 1.64·11-s + 0.810·13-s − 0.400·14-s − 1.10·16-s + 1.19·17-s + 1.32·19-s − 0.0559·20-s + 1.74·22-s − 1.08·23-s + 1/5·25-s − 0.859·26-s + 0.0472·28-s − 1.03·29-s + 1.36·31-s + 0.248·32-s − 1.27·34-s − 0.169·35-s − 1.67·37-s − 1.40·38-s − 0.415·40-s + 0.479·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 110 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 376 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 34 T + p^{3} T^{2} \) |
| 47 | \( 1 - 6 T + p^{3} T^{2} \) |
| 53 | \( 1 + 582 T + p^{3} T^{2} \) |
| 59 | \( 1 + 492 T + p^{3} T^{2} \) |
| 61 | \( 1 + 880 T + p^{3} T^{2} \) |
| 67 | \( 1 + 826 T + p^{3} T^{2} \) |
| 71 | \( 1 - 666 T + p^{3} T^{2} \) |
| 73 | \( 1 + 826 T + p^{3} T^{2} \) |
| 79 | \( 1 + 592 T + p^{3} T^{2} \) |
| 83 | \( 1 + 792 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1002 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1442 T + p^{3} T^{2} \) |
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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
−10.50330158489365955882557188522, −9.911381883658121032798486635180, −8.767931103556554796481045612733, −7.81628435365882593277923531714, −7.57444477318687232989936227330, −5.75829389662466937868867493509, −4.70691641096985925218101864826, −3.22395287623718505312847664315, −1.44446039907941357054892954820, 0,
1.44446039907941357054892954820, 3.22395287623718505312847664315, 4.70691641096985925218101864826, 5.75829389662466937868867493509, 7.57444477318687232989936227330, 7.81628435365882593277923531714, 8.767931103556554796481045612733, 9.911381883658121032798486635180, 10.50330158489365955882557188522
