| L(s) = 1 | + 2.82·5-s + 7-s + 2·11-s + 4.82·13-s + 4·17-s + 19-s − 0.828·23-s + 3.00·25-s − 7.65·29-s + 1.17·31-s + 2.82·35-s − 8.82·37-s + 2·41-s + 1.65·43-s + 10.4·47-s + 49-s + 6·53-s + 5.65·55-s + 4·59-s − 6·61-s + 13.6·65-s + 11.3·67-s + 13.6·71-s + 10·73-s + 2·77-s + 14.8·79-s − 15.6·83-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 0.377·7-s + 0.603·11-s + 1.33·13-s + 0.970·17-s + 0.229·19-s − 0.172·23-s + 0.600·25-s − 1.42·29-s + 0.210·31-s + 0.478·35-s − 1.45·37-s + 0.312·41-s + 0.252·43-s + 1.52·47-s + 0.142·49-s + 0.824·53-s + 0.762·55-s + 0.520·59-s − 0.768·61-s + 1.69·65-s + 1.38·67-s + 1.62·71-s + 1.17·73-s + 0.227·77-s + 1.66·79-s − 1.71·83-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
| \(L(1)\) |
\(\approx\) |
\(3.568924071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568924071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.68576171640838641716617819664, −6.90046403717253718927776892078, −6.24552459612375265420994281906, −5.54395625578144382551682015860, −5.30630480734659459205707917232, −4.00387119234400367666487843915, −3.57865061318861412133674264503, −2.43826742807181675283275948670, −1.66351920986472524263283125739, −0.991213546752919718324897353651,
0.991213546752919718324897353651, 1.66351920986472524263283125739, 2.43826742807181675283275948670, 3.57865061318861412133674264503, 4.00387119234400367666487843915, 5.30630480734659459205707917232, 5.54395625578144382551682015860, 6.24552459612375265420994281906, 6.90046403717253718927776892078, 7.68576171640838641716617819664
