| L(s) = 1 | + 5.21·2-s + 9·3-s − 4.77·4-s − 21.6·5-s + 46.9·6-s + 150.·7-s − 191.·8-s + 81·9-s − 113.·10-s + 527.·11-s − 42.9·12-s + 51.3·13-s + 783.·14-s − 194.·15-s − 848.·16-s − 968.·17-s + 422.·18-s + 2.27e3·19-s + 103.·20-s + 1.35e3·21-s + 2.75e3·22-s + 3.80e3·23-s − 1.72e3·24-s − 2.65e3·25-s + 268.·26-s + 729·27-s − 716.·28-s + ⋯ |
| L(s) = 1 | + 0.922·2-s + 0.577·3-s − 0.149·4-s − 0.387·5-s + 0.532·6-s + 1.15·7-s − 1.05·8-s + 0.333·9-s − 0.357·10-s + 1.31·11-s − 0.0861·12-s + 0.0843·13-s + 1.06·14-s − 0.223·15-s − 0.828·16-s − 0.812·17-s + 0.307·18-s + 1.44·19-s + 0.0577·20-s + 0.668·21-s + 1.21·22-s + 1.49·23-s − 0.611·24-s − 0.849·25-s + 0.0777·26-s + 0.192·27-s − 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.855464633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.855464633\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 - 5.21T + 32T^{2} \) |
| 5 | \( 1 + 21.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 150.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 527.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 51.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 968.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 513.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.13e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.70e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.53e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.20e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.31e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.24e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.91e3T + 8.58e9T^{2} \) |
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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
−11.79770329307389497252593652874, −11.29032003411233671622269902785, −9.538855733110085790133829023523, −8.770187004605072365878792716049, −7.71025252344735671382951893046, −6.39818648496672286499944513796, −4.92369195431987202584210881878, −4.18824586800093705256892171988, −2.98175022519739424329568927292, −1.19122934070908360872753927334,
1.19122934070908360872753927334, 2.98175022519739424329568927292, 4.18824586800093705256892171988, 4.92369195431987202584210881878, 6.39818648496672286499944513796, 7.71025252344735671382951893046, 8.770187004605072365878792716049, 9.538855733110085790133829023523, 11.29032003411233671622269902785, 11.79770329307389497252593652874
