| L(s) = 1 | − 1.44·2-s − 5.92·4-s + 4.95·5-s − 22.8·7-s + 20.0·8-s − 7.14·10-s + 59.8·11-s − 0.785·13-s + 32.9·14-s + 18.4·16-s + 118.·17-s − 22.5·19-s − 29.3·20-s − 86.2·22-s − 21.2·23-s − 100.·25-s + 1.13·26-s + 135.·28-s + 269.·29-s + 13.2·31-s − 187.·32-s − 170.·34-s − 113.·35-s + 306.·37-s + 32.5·38-s + 99.5·40-s + 214.·41-s + ⋯ |
| L(s) = 1 | − 0.509·2-s − 0.740·4-s + 0.443·5-s − 1.23·7-s + 0.887·8-s − 0.226·10-s + 1.63·11-s − 0.0167·13-s + 0.629·14-s + 0.287·16-s + 1.68·17-s − 0.272·19-s − 0.328·20-s − 0.835·22-s − 0.193·23-s − 0.803·25-s + 0.00854·26-s + 0.913·28-s + 1.72·29-s + 0.0767·31-s − 1.03·32-s − 0.859·34-s − 0.547·35-s + 1.36·37-s + 0.139·38-s + 0.393·40-s + 0.817·41-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
| \(L(2)\) |
\(\approx\) |
\(1.496988169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496988169\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.44T + 8T^{2} \) |
| 5 | \( 1 - 4.95T + 125T^{2} \) |
| 7 | \( 1 + 22.8T + 343T^{2} \) |
| 11 | \( 1 - 59.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.785T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 22.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 21.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 13.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 306.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 214.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 38.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 40.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 188.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 32.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 404.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 972.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 857.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 341.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 44.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 929.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 311.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 578.T + 9.12e5T^{2} \) |
| 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
−8.914572016453140966739956138312, −8.094672296483878927295819542648, −7.23685978341543365214428465542, −6.26141022892120914807907262509, −5.83650987508369469128507313795, −4.55925398934734022298651254966, −3.81711519100685983776572777881, −2.95400279979498400675371943995, −1.45130895262198744059275950867, −0.66811200705415957690385609501,
0.66811200705415957690385609501, 1.45130895262198744059275950867, 2.95400279979498400675371943995, 3.81711519100685983776572777881, 4.55925398934734022298651254966, 5.83650987508369469128507313795, 6.26141022892120914807907262509, 7.23685978341543365214428465542, 8.094672296483878927295819542648, 8.914572016453140966739956138312
