| L(s) = 1 | − 2.50·2-s + 4.25·4-s − 5-s + 3.78·7-s − 5.63·8-s + 2.50·10-s + 4.47·11-s + 5.13·13-s − 9.47·14-s + 5.59·16-s + 6.39·17-s − 7.47·19-s − 4.25·20-s − 11.1·22-s + 23-s + 25-s − 12.8·26-s + 16.1·28-s + 4.92·29-s − 1.25·31-s − 2.70·32-s − 15.9·34-s − 3.78·35-s + 6.54·37-s + 18.6·38-s + 5.63·40-s − 9.19·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 2.12·4-s − 0.447·5-s + 1.43·7-s − 1.99·8-s + 0.790·10-s + 1.34·11-s + 1.42·13-s − 2.53·14-s + 1.39·16-s + 1.55·17-s − 1.71·19-s − 0.951·20-s − 2.38·22-s + 0.208·23-s + 0.200·25-s − 2.51·26-s + 3.04·28-s + 0.914·29-s − 0.225·31-s − 0.478·32-s − 2.74·34-s − 0.640·35-s + 1.07·37-s + 3.03·38-s + 0.891·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9136370037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9136370037\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 - 6.39T + 17T^{2} \) |
| 19 | \( 1 + 7.47T + 19T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 - 6.54T + 37T^{2} \) |
| 41 | \( 1 + 9.19T + 41T^{2} \) |
| 43 | \( 1 + 3.17T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 + 8.98T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 8.33T + 71T^{2} \) |
| 73 | \( 1 + 1.13T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 3.06T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 4.42T + 97T^{2} \) |
| 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
−9.864286071096951083481560796846, −8.850896580751320863055417489817, −8.322013241680387449519295015188, −7.927016130413064271171768237938, −6.79365851185933202677802106522, −6.12182354194648317471966197534, −4.62569122328417452265991821032, −3.46498437054268546461572285335, −1.77850276655201039451453895012, −1.08371708456086627154792319917,
1.08371708456086627154792319917, 1.77850276655201039451453895012, 3.46498437054268546461572285335, 4.62569122328417452265991821032, 6.12182354194648317471966197534, 6.79365851185933202677802106522, 7.927016130413064271171768237938, 8.322013241680387449519295015188, 8.850896580751320863055417489817, 9.864286071096951083481560796846
