| L(s) = 1 | + 9·3-s − 11.6·5-s − 90.1·7-s + 81·9-s + 466.·11-s + 169·13-s − 104.·15-s + 1.39e3·17-s − 1.86e3·19-s − 811.·21-s + 4.16e3·23-s − 2.98e3·25-s + 729·27-s − 3.55e3·29-s + 1.95e3·31-s + 4.19e3·33-s + 1.04e3·35-s − 1.42e4·37-s + 1.52e3·39-s − 3.60e3·41-s − 1.70e3·43-s − 941.·45-s + 1.24e4·47-s − 8.67e3·49-s + 1.25e4·51-s + 2.41e3·53-s − 5.41e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.207·5-s − 0.695·7-s + 0.333·9-s + 1.16·11-s + 0.277·13-s − 0.120·15-s + 1.16·17-s − 1.18·19-s − 0.401·21-s + 1.64·23-s − 0.956·25-s + 0.192·27-s − 0.785·29-s + 0.364·31-s + 0.670·33-s + 0.144·35-s − 1.71·37-s + 0.160·39-s − 0.334·41-s − 0.140·43-s − 0.0693·45-s + 0.821·47-s − 0.516·49-s + 0.673·51-s + 0.117·53-s − 0.241·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.637279732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.637279732\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 - 169T \) |
| good | 5 | \( 1 + 11.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 90.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 466.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.39e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.16e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.41e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.66e4T + 8.58e9T^{2} \) |
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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
−9.689488232120836115931287947870, −8.992114994323481478423055003128, −8.212887225612001405273547203524, −7.10187771550064738202608344930, −6.44625700501869016340572445823, −5.26248305000633231775992636285, −3.89059007764416014080916947519, −3.37449818381322339712164425738, −1.99537174289807526117313582245, −0.77136800194101176246172206176,
0.77136800194101176246172206176, 1.99537174289807526117313582245, 3.37449818381322339712164425738, 3.89059007764416014080916947519, 5.26248305000633231775992636285, 6.44625700501869016340572445823, 7.10187771550064738202608344930, 8.212887225612001405273547203524, 8.992114994323481478423055003128, 9.689488232120836115931287947870
