| L(s) = 1 | − 16.8·2-s − 27·3-s + 155.·4-s + 189.·5-s + 454.·6-s + 904.·7-s − 469.·8-s + 729·9-s − 3.18e3·10-s − 4.29e3·11-s − 4.20e3·12-s + 1.02e3·13-s − 1.52e4·14-s − 5.10e3·15-s − 1.20e4·16-s + 1.81e4·17-s − 1.22e4·18-s − 2.36e4·19-s + 2.94e4·20-s − 2.44e4·21-s + 7.23e4·22-s − 7.59e4·23-s + 1.26e4·24-s − 4.23e4·25-s − 1.73e4·26-s − 1.96e4·27-s + 1.40e5·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.577·3-s + 1.21·4-s + 0.676·5-s + 0.859·6-s + 0.996·7-s − 0.324·8-s + 0.333·9-s − 1.00·10-s − 0.973·11-s − 0.703·12-s + 0.129·13-s − 1.48·14-s − 0.390·15-s − 0.734·16-s + 0.894·17-s − 0.496·18-s − 0.790·19-s + 0.823·20-s − 0.575·21-s + 1.44·22-s − 1.30·23-s + 0.187·24-s − 0.542·25-s − 0.193·26-s − 0.192·27-s + 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8034841557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8034841557\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 + 16.8T + 128T^{2} \) |
| 5 | \( 1 - 189.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 904.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.29e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.02e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.81e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.36e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.59e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.85e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 4.43e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.12e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.95e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.94e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.15e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.74e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.17e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.98e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.94e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.77e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.30e7T + 8.07e13T^{2} \) |
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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
−9.972789817015685054640231042108, −9.042402689302459556522906147621, −7.947478645363299165558011362329, −7.63894690612511325765789949856, −6.24190951930968080860883023406, −5.43116350296905605993217854629, −4.28049712399685421673436143311, −2.34132848185945675934454446510, −1.59902825502646208934072742572, −0.53059428263950929222264086193,
0.53059428263950929222264086193, 1.59902825502646208934072742572, 2.34132848185945675934454446510, 4.28049712399685421673436143311, 5.43116350296905605993217854629, 6.24190951930968080860883023406, 7.63894690612511325765789949856, 7.947478645363299165558011362329, 9.042402689302459556522906147621, 9.972789817015685054640231042108
