| L(s) = 1 | + 1.23·5-s + 7-s − 2·11-s − 4.47·13-s − 1.23·17-s − 19-s + 2·23-s − 3.47·25-s + 5.70·29-s − 1.52·31-s + 1.23·35-s + 8.47·37-s − 8.47·41-s + 8·43-s + 1.70·47-s + 49-s + 11.2·53-s − 2.47·55-s − 8.94·59-s − 13.4·61-s − 5.52·65-s − 1.52·67-s + 7.70·71-s + 12.4·73-s − 2·77-s + 4·79-s − 13.7·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s + 0.377·7-s − 0.603·11-s − 1.24·13-s − 0.299·17-s − 0.229·19-s + 0.417·23-s − 0.694·25-s + 1.05·29-s − 0.274·31-s + 0.208·35-s + 1.39·37-s − 1.32·41-s + 1.21·43-s + 0.249·47-s + 0.142·49-s + 1.54·53-s − 0.333·55-s − 1.16·59-s − 1.71·61-s − 0.685·65-s − 0.186·67-s + 0.914·71-s + 1.45·73-s − 0.227·77-s + 0.450·79-s − 1.50·83-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.41551329369254784495455900500, −6.67391760073726690808913027879, −5.94849896135790945206059990784, −5.22984499420197939245598446605, −4.70221612849787860038625425424, −3.92647562332410808656653822712, −2.70520558057069912964138036903, −2.36809958452679431388331325289, −1.28631455136984362766715061694, 0,
1.28631455136984362766715061694, 2.36809958452679431388331325289, 2.70520558057069912964138036903, 3.92647562332410808656653822712, 4.70221612849787860038625425424, 5.22984499420197939245598446605, 5.94849896135790945206059990784, 6.67391760073726690808913027879, 7.41551329369254784495455900500
