| L(s) = 1 | + 5-s + 0.331·7-s − 4.40·11-s + 6.98·13-s − 3.07·17-s + 7.55·19-s − 1.66·23-s + 25-s + 3.57·29-s − 5.64·31-s + 0.331·35-s + 7.64·37-s + 5.49·41-s + 7.07·43-s − 7.97·47-s − 6.88·49-s − 5.47·53-s − 4.40·55-s + 7.38·59-s + 0.592·61-s + 6.98·65-s − 5.22·67-s − 8.98·71-s − 8.05·73-s − 1.46·77-s + 10.9·79-s + 6.29·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.125·7-s − 1.32·11-s + 1.93·13-s − 0.744·17-s + 1.73·19-s − 0.347·23-s + 0.200·25-s + 0.663·29-s − 1.01·31-s + 0.0561·35-s + 1.25·37-s + 0.858·41-s + 1.07·43-s − 1.16·47-s − 0.984·49-s − 0.752·53-s − 0.594·55-s + 0.961·59-s + 0.0758·61-s + 0.865·65-s − 0.637·67-s − 1.06·71-s − 0.942·73-s − 0.166·77-s + 1.23·79-s + 0.690·83-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
| \(L(1)\) |
\(\approx\) |
\(2.319252484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319252484\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.331T + 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 - 6.98T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 - 7.55T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 + 5.64T + 31T^{2} \) |
| 37 | \( 1 - 7.64T + 37T^{2} \) |
| 41 | \( 1 - 5.49T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + 7.97T + 47T^{2} \) |
| 53 | \( 1 + 5.47T + 53T^{2} \) |
| 59 | \( 1 - 7.38T + 59T^{2} \) |
| 61 | \( 1 - 0.592T + 61T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 + 8.05T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 6.29T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.24T + 97T^{2} \) |
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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
−7.915981889565083572166218331575, −7.50629385869495944244885603102, −6.42076793873643625501271658952, −5.92580087899388782403044965996, −5.25187645704199186982455270215, −4.47353728382967180642034849876, −3.49058999990764495960044904378, −2.80886101622709067625655333676, −1.79694556849377185811927061473, −0.811831955513461632413192716327,
0.811831955513461632413192716327, 1.79694556849377185811927061473, 2.80886101622709067625655333676, 3.49058999990764495960044904378, 4.47353728382967180642034849876, 5.25187645704199186982455270215, 5.92580087899388782403044965996, 6.42076793873643625501271658952, 7.50629385869495944244885603102, 7.915981889565083572166218331575
