| L(s) = 1 | + 5-s + 2.73·7-s + 0.267·11-s + 1.46·13-s − 6.19·17-s − 19-s − 7.46·23-s + 25-s − 0.267·29-s + 2.46·31-s + 2.73·35-s − 7.26·37-s − 7.73·41-s − 6.19·43-s − 5.26·47-s + 0.464·49-s − 4.73·53-s + 0.267·55-s − 1.19·59-s − 14.9·61-s + 1.46·65-s + 0.535·67-s + 12.1·71-s − 1.26·73-s + 0.732·77-s + 8.53·79-s − 11.1·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.03·7-s + 0.0807·11-s + 0.406·13-s − 1.50·17-s − 0.229·19-s − 1.55·23-s + 0.200·25-s − 0.0497·29-s + 0.442·31-s + 0.461·35-s − 1.19·37-s − 1.20·41-s − 0.944·43-s − 0.768·47-s + 0.0663·49-s − 0.649·53-s + 0.0361·55-s − 0.155·59-s − 1.91·61-s + 0.181·65-s + 0.0654·67-s + 1.43·71-s − 0.148·73-s + 0.0834·77-s + 0.960·79-s − 1.22·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 0.267T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 + 0.267T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 + 4.73T + 53T^{2} \) |
| 59 | \( 1 + 1.19T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 0.535T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.86083549392838448023337091526, −6.75845509051040602925944311465, −6.39649182218527649792337962836, −5.46944590711805243924157247687, −4.76964738054951419822623910122, −4.16163787055474300653317896922, −3.17959618557660726393719770197, −2.00652562052619157910998324161, −1.62450688400419264140879573122, 0,
1.62450688400419264140879573122, 2.00652562052619157910998324161, 3.17959618557660726393719770197, 4.16163787055474300653317896922, 4.76964738054951419822623910122, 5.46944590711805243924157247687, 6.39649182218527649792337962836, 6.75845509051040602925944311465, 7.86083549392838448023337091526
