| L(s) = 1 | − 9·3-s + 81·5-s + 247·7-s + 81·9-s + 465·11-s − 694·13-s − 729·15-s + 543·17-s − 361·19-s − 2.22e3·21-s + 2.72e3·23-s + 3.43e3·25-s − 729·27-s + 342·29-s + 9.44e3·31-s − 4.18e3·33-s + 2.00e4·35-s + 1.30e4·37-s + 6.24e3·39-s − 1.62e4·41-s + 391·43-s + 6.56e3·45-s + 8.52e3·47-s + 4.42e4·49-s − 4.88e3·51-s − 1.01e4·53-s + 3.76e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.44·5-s + 1.90·7-s + 1/3·9-s + 1.15·11-s − 1.13·13-s − 0.836·15-s + 0.455·17-s − 0.229·19-s − 1.09·21-s + 1.07·23-s + 1.09·25-s − 0.192·27-s + 0.0755·29-s + 1.76·31-s − 0.668·33-s + 2.76·35-s + 1.57·37-s + 0.657·39-s − 1.51·41-s + 0.0322·43-s + 0.482·45-s + 0.562·47-s + 2.62·49-s − 0.263·51-s − 0.494·53-s + 1.67·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.918974585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.918974585\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 19 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 81 T + p^{5} T^{2} \) |
| 7 | \( 1 - 247 T + p^{5} T^{2} \) |
| 11 | \( 1 - 465 T + p^{5} T^{2} \) |
| 13 | \( 1 + 694 T + p^{5} T^{2} \) |
| 17 | \( 1 - 543 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2724 T + p^{5} T^{2} \) |
| 29 | \( 1 - 342 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9442 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13088 T + p^{5} T^{2} \) |
| 41 | \( 1 + 16272 T + p^{5} T^{2} \) |
| 43 | \( 1 - 391 T + p^{5} T^{2} \) |
| 47 | \( 1 - 8523 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10110 T + p^{5} T^{2} \) |
| 59 | \( 1 - 27144 T + p^{5} T^{2} \) |
| 61 | \( 1 + 48829 T + p^{5} T^{2} \) |
| 67 | \( 1 + 55448 T + p^{5} T^{2} \) |
| 71 | \( 1 + 43212 T + p^{5} T^{2} \) |
| 73 | \( 1 - 37685 T + p^{5} T^{2} \) |
| 79 | \( 1 - 78016 T + p^{5} T^{2} \) |
| 83 | \( 1 + 83892 T + p^{5} T^{2} \) |
| 89 | \( 1 - 25530 T + p^{5} T^{2} \) |
| 97 | \( 1 + 76378 T + p^{5} T^{2} \) |
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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
−9.468005375673293447307475521370, −8.594762486272356388130591756384, −7.60735883518352685272112501771, −6.66598856519307575520438079711, −5.81097808159646083307904660447, −4.96163308193912373592135422893, −4.45506629242849440065395549351, −2.62912296607469179625625719290, −1.62822006326517609454153967764, −1.02939418001019030725733821781,
1.02939418001019030725733821781, 1.62822006326517609454153967764, 2.62912296607469179625625719290, 4.45506629242849440065395549351, 4.96163308193912373592135422893, 5.81097808159646083307904660447, 6.66598856519307575520438079711, 7.60735883518352685272112501771, 8.594762486272356388130591756384, 9.468005375673293447307475521370
