| L(s) = 1 | + 1.66·5-s − 1.81·7-s + 3.97·11-s − 4.88·13-s + 3.09·17-s − 3.71·19-s − 2.52·23-s − 2.22·25-s − 4.16·29-s + 4.17·31-s − 3.02·35-s + 11.0·37-s + 0.0987·41-s − 2.04·43-s − 0.469·47-s − 3.69·49-s + 3.85·53-s + 6.62·55-s − 7.09·59-s − 0.640·61-s − 8.14·65-s + 1.93·67-s − 0.867·71-s − 6.51·73-s − 7.22·77-s − 9.82·79-s − 12.6·83-s + ⋯ |
| L(s) = 1 | + 0.745·5-s − 0.686·7-s + 1.19·11-s − 1.35·13-s + 0.750·17-s − 0.853·19-s − 0.525·23-s − 0.444·25-s − 0.772·29-s + 0.750·31-s − 0.512·35-s + 1.81·37-s + 0.0154·41-s − 0.312·43-s − 0.0684·47-s − 0.528·49-s + 0.530·53-s + 0.893·55-s − 0.924·59-s − 0.0819·61-s − 1.01·65-s + 0.236·67-s − 0.102·71-s − 0.762·73-s − 0.823·77-s − 1.10·79-s − 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 1.66T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 - 3.97T + 11T^{2} \) |
| 13 | \( 1 + 4.88T + 13T^{2} \) |
| 17 | \( 1 - 3.09T + 17T^{2} \) |
| 19 | \( 1 + 3.71T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 0.0987T + 41T^{2} \) |
| 43 | \( 1 + 2.04T + 43T^{2} \) |
| 47 | \( 1 + 0.469T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 + 0.640T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 + 0.867T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 9.81T + 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−7.39124468241116335086441025211, −6.51256729181908922924674171678, −6.16204388249107438811944362986, −5.45982575608307035939895007084, −4.52075841240327234931573540992, −3.91638074069008799804743133731, −2.94713444689768723966074668416, −2.22147712990958123545574776161, −1.32181619930289266960357592570, 0,
1.32181619930289266960357592570, 2.22147712990958123545574776161, 2.94713444689768723966074668416, 3.91638074069008799804743133731, 4.52075841240327234931573540992, 5.45982575608307035939895007084, 6.16204388249107438811944362986, 6.51256729181908922924674171678, 7.39124468241116335086441025211
