| L(s) = 1 | + 2.55·5-s + 3.62·7-s + 11-s − 13-s − 3.06·17-s − 3.62·19-s − 3.62·23-s + 1.51·25-s + 6.17·29-s + 4.55·31-s + 9.24·35-s + 4·37-s + 1.62·41-s + 6.17·43-s − 4.13·47-s + 6.10·49-s + 0.135·53-s + 2.55·55-s + 1.10·59-s + 1.86·61-s − 2.55·65-s + 11.7·67-s + 16.4·71-s + 16.7·73-s + 3.62·77-s + 9.06·79-s − 8.34·83-s + ⋯ |
| L(s) = 1 | + 1.14·5-s + 1.36·7-s + 0.301·11-s − 0.277·13-s − 0.744·17-s − 0.830·19-s − 0.754·23-s + 0.303·25-s + 1.14·29-s + 0.817·31-s + 1.56·35-s + 0.657·37-s + 0.253·41-s + 0.941·43-s − 0.603·47-s + 0.872·49-s + 0.0185·53-s + 0.344·55-s + 0.143·59-s + 0.238·61-s − 0.316·65-s + 1.44·67-s + 1.95·71-s + 1.95·73-s + 0.412·77-s + 1.02·79-s − 0.915·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.988877478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988877478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.55T + 5T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + 3.62T + 19T^{2} \) |
| 23 | \( 1 + 3.62T + 23T^{2} \) |
| 29 | \( 1 - 6.17T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 1.62T + 41T^{2} \) |
| 43 | \( 1 - 6.17T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 - 0.135T + 53T^{2} \) |
| 59 | \( 1 - 1.10T + 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.296716676530872940657235126932, −7.61864288967588876306392137559, −6.53745671562378012224860318232, −6.19605435427143029426213216014, −5.19841606923563294034036927242, −4.67894617104715915428786772118, −3.89192063175541775619500125467, −2.40861767783874913968270707842, −2.07303566262134601724895671578, −0.980972888358337944526607429518,
0.980972888358337944526607429518, 2.07303566262134601724895671578, 2.40861767783874913968270707842, 3.89192063175541775619500125467, 4.67894617104715915428786772118, 5.19841606923563294034036927242, 6.19605435427143029426213216014, 6.53745671562378012224860318232, 7.61864288967588876306392137559, 8.296716676530872940657235126932
