| L(s) = 1 | + 1.10·5-s − 2.52·7-s + 0.813·11-s + 5.10·13-s − 3.39·17-s − 3.10·19-s − 0.897·23-s − 3.78·25-s − 4.44·29-s + 8.96·31-s − 2.78·35-s + 2.08·37-s − 41-s − 9.07·43-s − 0.235·47-s − 0.627·49-s − 13.8·53-s + 0.897·55-s + 1.04·59-s + 1.91·61-s + 5.62·65-s + 10.0·67-s − 12.8·71-s + 4.75·73-s − 2.05·77-s + 15.8·79-s − 7.68·83-s + ⋯ |
| L(s) = 1 | + 0.493·5-s − 0.954·7-s + 0.245·11-s + 1.41·13-s − 0.822·17-s − 0.711·19-s − 0.187·23-s − 0.756·25-s − 0.824·29-s + 1.61·31-s − 0.470·35-s + 0.342·37-s − 0.156·41-s − 1.38·43-s − 0.0343·47-s − 0.0896·49-s − 1.90·53-s + 0.120·55-s + 0.136·59-s + 0.245·61-s + 0.697·65-s + 1.22·67-s − 1.52·71-s + 0.557·73-s − 0.234·77-s + 1.78·79-s − 0.843·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 - 0.813T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 + 3.39T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 + 0.897T + 23T^{2} \) |
| 29 | \( 1 + 4.44T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 0.235T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 - 1.91T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 4.75T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.72T + 97T^{2} \) |
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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
−7.86363167629277611870938531218, −6.66320794271881725350338621147, −6.41082475956436394719982671265, −5.84858393349400172838341175248, −4.80608348482543605059574107871, −3.94785971075696609077233768229, −3.30870642132600310749241101720, −2.31069977133058353840636799982, −1.38737517524335361383876198078, 0,
1.38737517524335361383876198078, 2.31069977133058353840636799982, 3.30870642132600310749241101720, 3.94785971075696609077233768229, 4.80608348482543605059574107871, 5.84858393349400172838341175248, 6.41082475956436394719982671265, 6.66320794271881725350338621147, 7.86363167629277611870938531218
