| L(s) = 1 | − 0.585·5-s − 3.41·7-s + 2·11-s + 2.82·13-s − 3.65·17-s + 5.65·19-s + 1.17·23-s − 4.65·25-s − 0.585·29-s − 4.58·31-s + 2·35-s − 9.65·37-s + 11.6·41-s − 1.65·43-s + 12.4·47-s + 4.65·49-s − 11.8·53-s − 1.17·55-s − 4·59-s − 9.65·61-s − 1.65·65-s + 8·67-s + 9.17·71-s − 1.65·73-s − 6.82·77-s − 5.75·79-s + 9.31·83-s + ⋯ |
| L(s) = 1 | − 0.261·5-s − 1.29·7-s + 0.603·11-s + 0.784·13-s − 0.886·17-s + 1.29·19-s + 0.244·23-s − 0.931·25-s − 0.108·29-s − 0.823·31-s + 0.338·35-s − 1.58·37-s + 1.82·41-s − 0.252·43-s + 1.82·47-s + 0.665·49-s − 1.63·53-s − 0.157·55-s − 0.520·59-s − 1.23·61-s − 0.205·65-s + 0.977·67-s + 1.08·71-s − 0.193·73-s − 0.778·77-s − 0.647·79-s + 1.02·83-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 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 0.585T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 9.65T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 + 1.65T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82558676650077260235677786110, −7.22745404745803242089590525532, −6.43581499984762172646290301150, −5.95494801008077829095745797460, −5.02831703732541069481056573312, −3.87363686950431121127608338070, −3.55151630954207785930339717133, −2.53779806018088467459933808297, −1.29473671637629931787079199586, 0,
1.29473671637629931787079199586, 2.53779806018088467459933808297, 3.55151630954207785930339717133, 3.87363686950431121127608338070, 5.02831703732541069481056573312, 5.95494801008077829095745797460, 6.43581499984762172646290301150, 7.22745404745803242089590525532, 7.82558676650077260235677786110
