| L(s) = 1 | + 0.797·2-s − 7.36·4-s + 7.16·5-s + 0.569·7-s − 12.2·8-s + 5.71·10-s + 72.6·11-s + 76.1·13-s + 0.453·14-s + 49.1·16-s − 76.7·17-s + 149.·19-s − 52.7·20-s + 57.9·22-s + 94.9·23-s − 73.6·25-s + 60.7·26-s − 4.19·28-s + 148.·29-s + 170.·31-s + 137.·32-s − 61.1·34-s + 4.07·35-s − 288.·37-s + 118.·38-s − 87.7·40-s − 144.·41-s + ⋯ |
| L(s) = 1 | + 0.281·2-s − 0.920·4-s + 0.640·5-s + 0.0307·7-s − 0.541·8-s + 0.180·10-s + 1.99·11-s + 1.62·13-s + 0.00866·14-s + 0.767·16-s − 1.09·17-s + 1.79·19-s − 0.589·20-s + 0.561·22-s + 0.860·23-s − 0.589·25-s + 0.458·26-s − 0.0282·28-s + 0.948·29-s + 0.988·31-s + 0.758·32-s − 0.308·34-s + 0.0196·35-s − 1.28·37-s + 0.507·38-s − 0.346·40-s − 0.551·41-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
| \(L(2)\) |
\(\approx\) |
\(3.222907898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.222907898\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.797T + 8T^{2} \) |
| 5 | \( 1 - 7.16T + 125T^{2} \) |
| 7 | \( 1 - 0.569T + 343T^{2} \) |
| 11 | \( 1 - 72.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 149.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 144.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 103.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 242.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 165.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 54.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 180.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 979.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 170.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 924.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 197.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 733.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 252.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.61e3T + 9.12e5T^{2} \) |
| 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
−8.878131531981004273687500389958, −8.248779142035675093820763034679, −6.87460811942722840747569969902, −6.33965527092105648354274434405, −5.55811927918719465489471550008, −4.65347085669789048646412382446, −3.82589771527567139726917140535, −3.17247079000719260908649911878, −1.53536013351021870441762899009, −0.895364863599652176560464205422,
0.895364863599652176560464205422, 1.53536013351021870441762899009, 3.17247079000719260908649911878, 3.82589771527567139726917140535, 4.65347085669789048646412382446, 5.55811927918719465489471550008, 6.33965527092105648354274434405, 6.87460811942722840747569969902, 8.248779142035675093820763034679, 8.878131531981004273687500389958
