| L(s) = 1 | − 19·5-s + 3·7-s − 2·11-s − 13·13-s − 77·17-s + 58·19-s + 76·23-s + 236·25-s + 6·29-s + 292·31-s − 57·35-s + 207·37-s − 240·41-s + 317·43-s − 375·47-s − 334·49-s + 692·53-s + 38·55-s + 214·59-s − 488·61-s + 247·65-s − 782·67-s − 1.05e3·71-s + 1.17e3·73-s − 6·77-s − 892·79-s + 704·83-s + ⋯ |
| L(s) = 1 | − 1.69·5-s + 0.161·7-s − 0.0548·11-s − 0.277·13-s − 1.09·17-s + 0.700·19-s + 0.689·23-s + 1.88·25-s + 0.0384·29-s + 1.69·31-s − 0.275·35-s + 0.919·37-s − 0.914·41-s + 1.12·43-s − 1.16·47-s − 0.973·49-s + 1.79·53-s + 0.0931·55-s + 0.472·59-s − 1.02·61-s + 0.471·65-s − 1.42·67-s − 1.76·71-s + 1.88·73-s − 0.00888·77-s − 1.27·79-s + 0.931·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 - 58 T + p^{3} T^{2} \) |
| 23 | \( 1 - 76 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 292 T + p^{3} T^{2} \) |
| 37 | \( 1 - 207 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 317 T + p^{3} T^{2} \) |
| 47 | \( 1 + 375 T + p^{3} T^{2} \) |
| 53 | \( 1 - 692 T + p^{3} T^{2} \) |
| 59 | \( 1 - 214 T + p^{3} T^{2} \) |
| 61 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 782 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1057 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1174 T + p^{3} T^{2} \) |
| 79 | \( 1 + 892 T + p^{3} T^{2} \) |
| 83 | \( 1 - 704 T + p^{3} T^{2} \) |
| 89 | \( 1 + 6 T + p^{3} T^{2} \) |
| 97 | \( 1 - 830 T + p^{3} T^{2} \) |
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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
−8.406427243031185353604584066173, −7.68735335528294431719381048601, −7.10056189758865175860079837360, −6.23112227756067762475575530810, −4.88326018423582523984022150172, −4.43903349299717433789553104252, −3.46177645019298065928045377141, −2.60504169046124223934392300468, −1.02800526461543736746214770921, 0,
1.02800526461543736746214770921, 2.60504169046124223934392300468, 3.46177645019298065928045377141, 4.43903349299717433789553104252, 4.88326018423582523984022150172, 6.23112227756067762475575530810, 7.10056189758865175860079837360, 7.68735335528294431719381048601, 8.406427243031185353604584066173
