| L(s) = 1 | − 4·5-s + 10·7-s − 4·11-s + 26·13-s − 14·17-s − 8·19-s − 148·23-s − 109·25-s − 72·29-s + 18·31-s − 40·35-s + 262·37-s + 378·41-s + 432·43-s − 148·47-s − 243·49-s − 360·53-s + 16·55-s − 428·59-s − 442·61-s − 104·65-s + 692·67-s − 540·71-s − 1.01e3·73-s − 40·77-s + 386·79-s + 108·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 0.539·7-s − 0.109·11-s + 0.554·13-s − 0.199·17-s − 0.0965·19-s − 1.34·23-s − 0.871·25-s − 0.461·29-s + 0.104·31-s − 0.193·35-s + 1.16·37-s + 1.43·41-s + 1.53·43-s − 0.459·47-s − 0.708·49-s − 0.933·53-s + 0.0392·55-s − 0.944·59-s − 0.927·61-s − 0.198·65-s + 1.26·67-s − 0.902·71-s − 1.63·73-s − 0.0592·77-s + 0.549·79-s + 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 8 T + p^{3} T^{2} \) |
| 23 | \( 1 + 148 T + p^{3} T^{2} \) |
| 29 | \( 1 + 72 T + p^{3} T^{2} \) |
| 31 | \( 1 - 18 T + p^{3} T^{2} \) |
| 37 | \( 1 - 262 T + p^{3} T^{2} \) |
| 41 | \( 1 - 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 432 T + p^{3} T^{2} \) |
| 47 | \( 1 + 148 T + p^{3} T^{2} \) |
| 53 | \( 1 + 360 T + p^{3} T^{2} \) |
| 59 | \( 1 + 428 T + p^{3} T^{2} \) |
| 61 | \( 1 + 442 T + p^{3} T^{2} \) |
| 67 | \( 1 - 692 T + p^{3} T^{2} \) |
| 71 | \( 1 + 540 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1018 T + p^{3} T^{2} \) |
| 79 | \( 1 - 386 T + p^{3} T^{2} \) |
| 83 | \( 1 - 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 382 T + p^{3} T^{2} \) |
| 97 | \( 1 - 298 T + p^{3} T^{2} \) |
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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
−9.034695844081450272034064768225, −7.909519344457934495131049586572, −7.73420167908869558485880031621, −6.36182885949833734033414359105, −5.71918091315751831885640077013, −4.50528230456065566546032222487, −3.86190189187467988495782260599, −2.56162898426669400790295880001, −1.41834745374186139121725671442, 0,
1.41834745374186139121725671442, 2.56162898426669400790295880001, 3.86190189187467988495782260599, 4.50528230456065566546032222487, 5.71918091315751831885640077013, 6.36182885949833734033414359105, 7.73420167908869558485880031621, 7.909519344457934495131049586572, 9.034695844081450272034064768225
