| L(s) = 1 | − 5.46e3·3-s − 3.46e5·5-s − 4.59e5·7-s + 1.55e7·9-s + 1.11e8·11-s + 8.87e7·13-s + 1.89e9·15-s − 6.77e8·17-s + 8.93e8·19-s + 2.51e9·21-s + 1.61e10·23-s + 8.97e10·25-s − 6.48e9·27-s + 7.38e9·29-s + 2.05e11·31-s − 6.11e11·33-s + 1.59e11·35-s − 7.83e11·37-s − 4.85e11·39-s − 4.30e10·41-s + 3.42e12·43-s − 5.38e12·45-s − 3.76e12·47-s − 4.53e12·49-s + 3.70e12·51-s + 4.66e12·53-s − 3.88e13·55-s + ⋯ |
| L(s) = 1 | − 1.44·3-s − 1.98·5-s − 0.210·7-s + 1.08·9-s + 1.73·11-s + 0.392·13-s + 2.86·15-s − 0.400·17-s + 0.229·19-s + 0.304·21-s + 0.989·23-s + 2.94·25-s − 0.119·27-s + 0.0794·29-s + 1.33·31-s − 2.49·33-s + 0.418·35-s − 1.35·37-s − 0.566·39-s − 0.0345·41-s + 1.91·43-s − 2.14·45-s − 1.08·47-s − 0.955·49-s + 0.578·51-s + 0.545·53-s − 3.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.8755248584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8755248584\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 8.93e8T \) |
| good | 3 | \( 1 + 5.46e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 3.46e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 4.59e5T + 4.74e12T^{2} \) |
| 11 | \( 1 - 1.11e8T + 4.17e15T^{2} \) |
| 13 | \( 1 - 8.87e7T + 5.11e16T^{2} \) |
| 17 | \( 1 + 6.77e8T + 2.86e18T^{2} \) |
| 23 | \( 1 - 1.61e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 7.38e9T + 8.62e21T^{2} \) |
| 31 | \( 1 - 2.05e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 7.83e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 4.30e10T + 1.55e24T^{2} \) |
| 43 | \( 1 - 3.42e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 3.76e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 4.66e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.61e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 9.66e12T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.63e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.27e14T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.50e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.83e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 9.72e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 2.44e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.86e14T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84617716210737883997197155185, −9.238004358042689293687602658498, −8.177347677061823711426755931126, −6.91891256747625464399105689622, −6.46015131145976182246491771511, −4.95358981927027932392191908740, −4.16250711772338631987648522429, −3.30505464030981303216462945208, −1.12934219349722933090077089900, −0.52328342891418754492040620658,
0.52328342891418754492040620658, 1.12934219349722933090077089900, 3.30505464030981303216462945208, 4.16250711772338631987648522429, 4.95358981927027932392191908740, 6.46015131145976182246491771511, 6.91891256747625464399105689622, 8.177347677061823711426755931126, 9.238004358042689293687602658498, 10.84617716210737883997197155185
