| L(s) = 1 | + 1.35e14·2-s + 8.34e27·4-s − 3.52e32·5-s − 7.46e38·7-s − 2.10e41·8-s − 4.75e46·10-s + 9.09e47·11-s − 6.25e51·13-s − 1.00e53·14-s − 1.11e56·16-s − 2.83e57·17-s − 8.20e57·19-s − 2.94e60·20-s + 1.22e62·22-s + 2.05e63·23-s + 2.30e64·25-s − 8.44e65·26-s − 6.23e66·28-s − 1.94e68·29-s + 8.59e68·31-s − 1.29e70·32-s − 3.83e71·34-s + 2.62e71·35-s + 9.63e71·37-s − 1.10e72·38-s + 7.40e73·40-s − 1.11e75·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.842·4-s − 1.10·5-s − 0.376·7-s − 0.213·8-s − 1.50·10-s + 0.342·11-s − 0.994·13-s − 0.511·14-s − 1.13·16-s − 1.72·17-s − 0.0283·19-s − 0.934·20-s + 0.464·22-s + 0.984·23-s + 0.228·25-s − 1.35·26-s − 0.317·28-s − 1.94·29-s + 0.385·31-s − 1.32·32-s − 2.34·34-s + 0.417·35-s + 0.115·37-s − 0.0384·38-s + 0.236·40-s − 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(94-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+93/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(47)\) |
\(\approx\) |
\(0.2648948194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2648948194\) |
| \(L(\frac{95}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.35e14T + 9.90e27T^{2} \) |
| 5 | \( 1 + 3.52e32T + 1.00e65T^{2} \) |
| 7 | \( 1 + 7.46e38T + 3.92e78T^{2} \) |
| 11 | \( 1 - 9.09e47T + 7.07e96T^{2} \) |
| 13 | \( 1 + 6.25e51T + 3.95e103T^{2} \) |
| 17 | \( 1 + 2.83e57T + 2.70e114T^{2} \) |
| 19 | \( 1 + 8.20e57T + 8.39e118T^{2} \) |
| 23 | \( 1 - 2.05e63T + 4.37e126T^{2} \) |
| 29 | \( 1 + 1.94e68T + 1.00e136T^{2} \) |
| 31 | \( 1 - 8.59e68T + 4.97e138T^{2} \) |
| 37 | \( 1 - 9.63e71T + 6.96e145T^{2} \) |
| 41 | \( 1 + 1.11e75T + 9.74e149T^{2} \) |
| 43 | \( 1 + 1.49e76T + 8.17e151T^{2} \) |
| 47 | \( 1 - 6.74e75T + 3.19e155T^{2} \) |
| 53 | \( 1 + 2.18e80T + 2.27e160T^{2} \) |
| 59 | \( 1 - 4.18e82T + 4.88e164T^{2} \) |
| 61 | \( 1 - 3.83e82T + 1.08e166T^{2} \) |
| 67 | \( 1 - 2.76e84T + 6.68e169T^{2} \) |
| 71 | \( 1 + 1.96e86T + 1.46e172T^{2} \) |
| 73 | \( 1 + 1.54e86T + 1.94e173T^{2} \) |
| 79 | \( 1 - 8.28e87T + 3.01e176T^{2} \) |
| 83 | \( 1 + 1.06e89T + 2.98e178T^{2} \) |
| 89 | \( 1 - 1.04e90T + 1.96e181T^{2} \) |
| 97 | \( 1 + 3.92e92T + 5.88e184T^{2} \) |
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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
−9.575715844968978178093476569173, −8.526065290173751959591494809213, −7.18299519632421296845018488588, −6.54744195179035904615299363484, −5.23721308058965012381950968403, −4.48420013334885613177871916639, −3.72995497728774943848326120611, −2.93560174929661518040299483199, −1.88234515821817780701991069917, −0.13550109346700945228602016179,
0.13550109346700945228602016179, 1.88234515821817780701991069917, 2.93560174929661518040299483199, 3.72995497728774943848326120611, 4.48420013334885613177871916639, 5.23721308058965012381950968403, 6.54744195179035904615299363484, 7.18299519632421296845018488588, 8.526065290173751959591494809213, 9.575715844968978178093476569173
