| L(s) = 1 | + 0.449·5-s − 2.04·7-s + 3.46·11-s − 4.79·25-s − 9.34·29-s − 3.60·31-s − 0.921·35-s − 2.79·49-s − 7.55·53-s + 1.55·55-s + 11.3·59-s + 9.79·73-s − 7.10·77-s + 17.4·79-s − 17.3·83-s − 2·97-s − 16.4·101-s + 0.492·103-s + 11.3·107-s + ⋯ |
| L(s) = 1 | + 0.201·5-s − 0.774·7-s + 1.04·11-s − 0.959·25-s − 1.73·29-s − 0.647·31-s − 0.155·35-s − 0.399·49-s − 1.03·53-s + 0.209·55-s + 1.47·59-s + 1.14·73-s − 0.809·77-s + 1.96·79-s − 1.90·83-s − 0.203·97-s − 1.63·101-s + 0.0485·103-s + 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.449T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.914176715350918566486585646486, −7.15784159993987680250330505594, −6.47905504625610074695028828004, −5.86814817996398070544519789692, −5.08704173172540960328299800431, −3.91347802080894522581071908547, −3.57082452740568230864889547947, −2.37320861918770211587373865325, −1.43598925917352209943207387673, 0,
1.43598925917352209943207387673, 2.37320861918770211587373865325, 3.57082452740568230864889547947, 3.91347802080894522581071908547, 5.08704173172540960328299800431, 5.86814817996398070544519789692, 6.47905504625610074695028828004, 7.15784159993987680250330505594, 7.914176715350918566486585646486
