| L(s) = 1 | + 84.8·5-s − 49·7-s + 634.·11-s + 895.·13-s − 2.05e3·17-s + 2.45e3·19-s + 569.·23-s + 4.07e3·25-s − 1.47e3·29-s − 2.00e3·31-s − 4.15e3·35-s + 4.86e3·37-s + 1.72e4·41-s − 1.54e4·43-s − 5.00e3·47-s + 2.40e3·49-s + 1.95e4·53-s + 5.37e4·55-s − 1.45e4·59-s + 3.57e3·61-s + 7.59e4·65-s − 4.14e4·67-s − 9.24e3·71-s − 4.13e4·73-s − 3.10e4·77-s − 3.79e4·79-s − 7.92e4·83-s + ⋯ |
| L(s) = 1 | + 1.51·5-s − 0.377·7-s + 1.58·11-s + 1.46·13-s − 1.72·17-s + 1.55·19-s + 0.224·23-s + 1.30·25-s − 0.324·29-s − 0.375·31-s − 0.573·35-s + 0.583·37-s + 1.60·41-s − 1.27·43-s − 0.330·47-s + 0.142·49-s + 0.956·53-s + 2.39·55-s − 0.542·59-s + 0.122·61-s + 2.22·65-s − 1.12·67-s − 0.217·71-s − 0.908·73-s − 0.597·77-s − 0.684·79-s − 1.26·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.522982644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.522982644\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 49T \) |
| good | 5 | \( 1 - 84.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 634.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 895.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 569.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.86e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.54e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.57e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.24e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.92e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.25e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.75e5T + 8.58e9T^{2} \) |
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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
−9.945738589340981870810183974895, −9.173255791469530684378621650108, −8.797247173010427854038667277312, −7.12009237251881055079286538950, −6.28459204322698475909693815126, −5.77751707637073823009042271985, −4.38120249148053331463569634479, −3.23339235025445768334003566736, −1.88538546710140790162252750431, −1.03336101593538459224678965708,
1.03336101593538459224678965708, 1.88538546710140790162252750431, 3.23339235025445768334003566736, 4.38120249148053331463569634479, 5.77751707637073823009042271985, 6.28459204322698475909693815126, 7.12009237251881055079286538950, 8.797247173010427854038667277312, 9.173255791469530684378621650108, 9.945738589340981870810183974895
