| L(s) = 1 | + 2.60·2-s − 1.23·4-s + 11.0·5-s − 16.8·7-s − 24.0·8-s + 28.7·10-s + 0.0148·11-s − 3.98·13-s − 43.8·14-s − 52.5·16-s − 111.·17-s − 51.9·19-s − 13.7·20-s + 0.0384·22-s − 134.·23-s − 2.59·25-s − 10.3·26-s + 20.8·28-s + 114.·29-s + 213.·31-s + 55.5·32-s − 289.·34-s − 186.·35-s + 391.·37-s − 135.·38-s − 265.·40-s + 345.·41-s + ⋯ |
| L(s) = 1 | + 0.919·2-s − 0.154·4-s + 0.989·5-s − 0.910·7-s − 1.06·8-s + 0.909·10-s + 0.000405·11-s − 0.0849·13-s − 0.836·14-s − 0.821·16-s − 1.58·17-s − 0.627·19-s − 0.153·20-s + 0.000373·22-s − 1.21·23-s − 0.0207·25-s − 0.0780·26-s + 0.140·28-s + 0.733·29-s + 1.23·31-s + 0.306·32-s − 1.46·34-s − 0.900·35-s + 1.74·37-s − 0.576·38-s − 1.05·40-s + 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.405427123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.405427123\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 2.60T + 8T^{2} \) |
| 5 | \( 1 - 11.0T + 125T^{2} \) |
| 7 | \( 1 + 16.8T + 343T^{2} \) |
| 11 | \( 1 - 0.0148T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.98T + 2.19e3T^{2} \) |
| 17 | \( 1 + 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 442.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 222.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 665.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 584.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 21.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 427.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 66.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 342.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 884.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 703.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 960.T + 9.12e5T^{2} \) |
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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
−8.940387466784010872162342097989, −7.999006912676631081605049497246, −6.70944635092729519649191525101, −6.13779538124498715127650918689, −5.75804787289134735134744559471, −4.46068859516083066255923476894, −4.14405274695232929920260782933, −2.78164323418875459765797365111, −2.27607883972317755310143498817, −0.58279609716786469954730851078,
0.58279609716786469954730851078, 2.27607883972317755310143498817, 2.78164323418875459765797365111, 4.14405274695232929920260782933, 4.46068859516083066255923476894, 5.75804787289134735134744559471, 6.13779538124498715127650918689, 6.70944635092729519649191525101, 7.999006912676631081605049497246, 8.940387466784010872162342097989
