| L(s) = 1 | + 5.16·2-s − 3·3-s + 18.6·4-s + 13.7·5-s − 15.4·6-s + 32.4·7-s + 55.1·8-s + 9·9-s + 71.1·10-s − 43.8·11-s − 56.0·12-s + 45.9·13-s + 167.·14-s − 41.3·15-s + 135.·16-s − 49.2·17-s + 46.4·18-s − 121.·19-s + 257.·20-s − 97.2·21-s − 226.·22-s − 164.·23-s − 165.·24-s + 64.5·25-s + 237.·26-s − 27·27-s + 605.·28-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.577·3-s + 2.33·4-s + 1.23·5-s − 1.05·6-s + 1.75·7-s + 2.43·8-s + 0.333·9-s + 2.24·10-s − 1.20·11-s − 1.34·12-s + 0.981·13-s + 3.19·14-s − 0.710·15-s + 2.11·16-s − 0.702·17-s + 0.608·18-s − 1.47·19-s + 2.87·20-s − 1.01·21-s − 2.19·22-s − 1.48·23-s − 1.40·24-s + 0.516·25-s + 1.79·26-s − 0.192·27-s + 4.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.943400186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.943400186\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + 157T \) |
| good | 2 | \( 1 - 5.16T + 8T^{2} \) |
| 5 | \( 1 - 13.7T + 125T^{2} \) |
| 7 | \( 1 - 32.4T + 343T^{2} \) |
| 11 | \( 1 + 43.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 298.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 438.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 78.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 23.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 68.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 83.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 569.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 368.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 534.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 960.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 773.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 701.T + 9.12e5T^{2} \) |
| 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
−10.77833342099851121196873111867, −10.45410256730041292904075199604, −8.598918947425713624582925088375, −7.55445808499064941618341633345, −6.19244014061443333091949346837, −5.82806905046980798007802116256, −4.84990337386085903314561044916, −4.24360825923448451130101013053, −2.40184162305548944765176886022, −1.71924619203400905792874502724,
1.71924619203400905792874502724, 2.40184162305548944765176886022, 4.24360825923448451130101013053, 4.84990337386085903314561044916, 5.82806905046980798007802116256, 6.19244014061443333091949346837, 7.55445808499064941618341633345, 8.598918947425713624582925088375, 10.45410256730041292904075199604, 10.77833342099851121196873111867
