| L(s) = 1 | + 0.594·2-s − 7.64·4-s + 5.61·5-s − 8.56·7-s − 9.30·8-s + 3.34·10-s − 46.3·11-s − 29.3·13-s − 5.09·14-s + 55.6·16-s + 123.·17-s − 53.5·19-s − 42.9·20-s − 27.5·22-s + 172.·23-s − 93.4·25-s − 17.4·26-s + 65.5·28-s + 128.·29-s − 40.2·31-s + 107.·32-s + 73.2·34-s − 48.1·35-s + 398.·37-s − 31.8·38-s − 52.2·40-s − 26.9·41-s + ⋯ |
| L(s) = 1 | + 0.210·2-s − 0.955·4-s + 0.502·5-s − 0.462·7-s − 0.411·8-s + 0.105·10-s − 1.27·11-s − 0.625·13-s − 0.0972·14-s + 0.869·16-s + 1.75·17-s − 0.646·19-s − 0.480·20-s − 0.267·22-s + 1.56·23-s − 0.747·25-s − 0.131·26-s + 0.442·28-s + 0.822·29-s − 0.233·31-s + 0.593·32-s + 0.369·34-s − 0.232·35-s + 1.76·37-s − 0.135·38-s − 0.206·40-s − 0.102·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.594T + 8T^{2} \) |
| 5 | \( 1 - 5.61T + 125T^{2} \) |
| 7 | \( 1 + 8.56T + 343T^{2} \) |
| 11 | \( 1 + 46.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 53.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 398.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 26.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 62.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 207.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 515.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 30.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 27.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 593.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 804.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 283.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 284.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 689.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.81e3T + 9.12e5T^{2} \) |
| 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
−8.212193067023486417685995364945, −7.79183402252838658281555473597, −6.69277666101005207618095834929, −5.67340257563826611090502538352, −5.24363148801853814017512694456, −4.39485920057833684547292118939, −3.26013440495905137240406753129, −2.59963488802166599615047935854, −1.07899753031850784916911384789, 0,
1.07899753031850784916911384789, 2.59963488802166599615047935854, 3.26013440495905137240406753129, 4.39485920057833684547292118939, 5.24363148801853814017512694456, 5.67340257563826611090502538352, 6.69277666101005207618095834929, 7.79183402252838658281555473597, 8.212193067023486417685995364945
