| L(s) = 1 | − 2.82·5-s + 7-s + 2·11-s − 0.828·13-s + 4·17-s + 19-s + 4.82·23-s + 3.00·25-s + 3.65·29-s + 6.82·31-s − 2.82·35-s − 3.17·37-s + 2·41-s − 9.65·43-s − 6.48·47-s + 49-s + 6·53-s − 5.65·55-s + 4·59-s − 6·61-s + 2.34·65-s − 11.3·67-s + 2.34·71-s + 10·73-s + 2·77-s + 9.17·79-s − 4.34·83-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 0.377·7-s + 0.603·11-s − 0.229·13-s + 0.970·17-s + 0.229·19-s + 1.00·23-s + 0.600·25-s + 0.679·29-s + 1.22·31-s − 0.478·35-s − 0.521·37-s + 0.312·41-s − 1.47·43-s − 0.945·47-s + 0.142·49-s + 0.824·53-s − 0.762·55-s + 0.520·59-s − 0.768·61-s + 0.290·65-s − 1.38·67-s + 0.278·71-s + 1.17·73-s + 0.227·77-s + 1.03·79-s − 0.476·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727486238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727486238\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.17T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 4.34T + 83T^{2} \) |
| 89 | \( 1 + 0.343T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−7.88142458468792024348663577290, −6.98522082407248989967374505284, −6.57575646943681941631616328515, −5.49632166475765640179097435652, −4.84382735532714466020125342564, −4.20470924509996335250549213574, −3.43549806058504962570421027651, −2.84853032749248941539428549305, −1.53898920487762721399694196299, −0.66667112308191374579033810868,
0.66667112308191374579033810868, 1.53898920487762721399694196299, 2.84853032749248941539428549305, 3.43549806058504962570421027651, 4.20470924509996335250549213574, 4.84382735532714466020125342564, 5.49632166475765640179097435652, 6.57575646943681941631616328515, 6.98522082407248989967374505284, 7.88142458468792024348663577290
