| L(s) = 1 | + i·2-s − 4-s + (−2 + i)5-s + 2i·7-s − i·8-s + (−1 − 2i)10-s − 4·11-s + 2i·13-s − 2·14-s + 16-s + 6i·17-s − 4·19-s + (2 − i)20-s − 4i·22-s + (3 − 4i)25-s − 2·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 + 0.447i)5-s + 0.755i·7-s − 0.353i·8-s + (−0.316 − 0.632i)10-s − 1.20·11-s + 0.554i·13-s − 0.534·14-s + 0.250·16-s + 1.45i·17-s − 0.917·19-s + (0.447 − 0.223i)20-s − 0.852i·22-s + (0.600 − 0.800i)25-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
| 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
−8.443526402507529929652977577658, −7.965638331492602594835219951997, −7.23148342103075596824245402557, −6.34509135702991782263506703902, −5.77950216086982662610895662520, −4.75826879822287049803841803334, −4.04667351684374894480029337344, −3.05772097400767880301475517534, −1.99394870482206160954558242456, 0,
0.910568133019208750890255300881, 2.42254186938460700923420777254, 3.26090820264172181600487675934, 4.16772082755893059658766073818, 4.86035980561966776626894778249, 5.54964793270024469776028401590, 6.90452554282023485527970776922, 7.60774388664803344006581988822, 8.160777243028091177222112027472
