| L(s) = 1 | + 2.58·5-s + 7-s − 4.04·11-s − 0.190·13-s − 6.82·17-s − 1.34·19-s + 0.240·23-s + 1.65·25-s + 0.190·29-s + 2.46·31-s + 2.58·35-s + 10.4·37-s − 2.77·41-s − 11.4·43-s + 4.29·47-s + 49-s + 9.26·53-s − 10.4·55-s − 12.9·59-s − 13.6·61-s − 0.490·65-s − 0.871·67-s − 11.0·71-s + 12.5·73-s − 4.04·77-s − 15.0·79-s − 4.19·83-s + ⋯ |
| L(s) = 1 | + 1.15·5-s + 0.377·7-s − 1.22·11-s − 0.0527·13-s − 1.65·17-s − 0.307·19-s + 0.0500·23-s + 0.331·25-s + 0.0353·29-s + 0.443·31-s + 0.436·35-s + 1.71·37-s − 0.432·41-s − 1.74·43-s + 0.625·47-s + 0.142·49-s + 1.27·53-s − 1.40·55-s − 1.68·59-s − 1.74·61-s − 0.0608·65-s − 0.106·67-s − 1.31·71-s + 1.46·73-s − 0.461·77-s − 1.69·79-s − 0.459·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 2.58T + 5T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 0.190T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 - 0.240T + 23T^{2} \) |
| 29 | \( 1 - 0.190T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.77T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 - 9.26T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 0.871T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 4.19T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
| 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
−7.76954723474778312947211965505, −6.94226869461527589511604086296, −6.21184848298912722960328347088, −5.64517033927695183228115278100, −4.82476686547887270950514035331, −4.29547302694165690376662993102, −2.90652485638411626486290873628, −2.33724498217187129317758827266, −1.52975403692924058925749713638, 0,
1.52975403692924058925749713638, 2.33724498217187129317758827266, 2.90652485638411626486290873628, 4.29547302694165690376662993102, 4.82476686547887270950514035331, 5.64517033927695183228115278100, 6.21184848298912722960328347088, 6.94226869461527589511604086296, 7.76954723474778312947211965505
