| L(s) = 1 | − 12·5-s + 7·7-s + 60·11-s − 79·13-s + 108·17-s − 11·19-s − 132·23-s + 19·25-s − 96·29-s − 20·31-s − 84·35-s − 169·37-s − 192·41-s − 488·43-s + 204·47-s − 294·49-s − 360·53-s − 720·55-s + 156·59-s + 83·61-s + 948·65-s − 47·67-s + 216·71-s − 511·73-s + 420·77-s + 529·79-s − 1.12e3·83-s + ⋯ |
| L(s) = 1 | − 1.07·5-s + 0.377·7-s + 1.64·11-s − 1.68·13-s + 1.54·17-s − 0.132·19-s − 1.19·23-s + 0.151·25-s − 0.614·29-s − 0.115·31-s − 0.405·35-s − 0.750·37-s − 0.731·41-s − 1.73·43-s + 0.633·47-s − 6/7·49-s − 0.933·53-s − 1.76·55-s + 0.344·59-s + 0.174·61-s + 1.80·65-s − 0.0857·67-s + 0.361·71-s − 0.819·73-s + 0.621·77-s + 0.753·79-s − 1.49·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - p T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 + 79 T + p^{3} T^{2} \) |
| 17 | \( 1 - 108 T + p^{3} T^{2} \) |
| 19 | \( 1 + 11 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 96 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 + 169 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 + 488 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 + 360 T + p^{3} T^{2} \) |
| 59 | \( 1 - 156 T + p^{3} T^{2} \) |
| 61 | \( 1 - 83 T + p^{3} T^{2} \) |
| 67 | \( 1 + 47 T + p^{3} T^{2} \) |
| 71 | \( 1 - 216 T + p^{3} T^{2} \) |
| 73 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 529 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1128 T + p^{3} T^{2} \) |
| 89 | \( 1 + 36 T + p^{3} T^{2} \) |
| 97 | \( 1 - 605 T + p^{3} T^{2} \) |
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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
−10.16705554700907155417047874069, −9.475250845116054455416640721874, −8.274775552333309641106801205794, −7.58114725371880303729728649127, −6.70735054963997586804418532818, −5.34930863499319807939819524582, −4.25037341599601967426124841902, −3.38380493414543739187800147769, −1.65125127733110365608851543167, 0,
1.65125127733110365608851543167, 3.38380493414543739187800147769, 4.25037341599601967426124841902, 5.34930863499319807939819524582, 6.70735054963997586804418532818, 7.58114725371880303729728649127, 8.274775552333309641106801205794, 9.475250845116054455416640721874, 10.16705554700907155417047874069
