L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.64i·7-s + 2.82·8-s + 5.31i·11-s + 0.354i·13-s − 3.74i·14-s + 4.00·16-s − 31.5·17-s + 28.2·19-s + 7.52i·22-s − 25.0·23-s + 0.500i·26-s − 5.29i·28-s − 3.16i·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.377i·7-s + 0.353·8-s + 0.483i·11-s + 0.0272i·13-s − 0.267i·14-s + 0.250·16-s − 1.85·17-s + 1.48·19-s + 0.341i·22-s − 1.08·23-s + 0.0192i·26-s − 0.188i·28-s − 0.109i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.839136764\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.839136764\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 11 | \( 1 - 5.31iT - 121T^{2} \) |
| 13 | \( 1 - 0.354iT - 169T^{2} \) |
| 17 | \( 1 + 31.5T + 289T^{2} \) |
| 19 | \( 1 - 28.2T + 361T^{2} \) |
| 23 | \( 1 + 25.0T + 529T^{2} \) |
| 29 | \( 1 + 3.16iT - 841T^{2} \) |
| 31 | \( 1 - 25.1T + 961T^{2} \) |
| 37 | \( 1 - 41.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 70.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 84.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 46.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 41.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 56.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 86.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 128.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 142. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 133. iT - 9.40e3T^{2} \) |
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
−8.522886912433969095965350138380, −7.75463373754080815528053253149, −7.00223126819548152811871690363, −6.39248637716692879428464238363, −5.55220967109087626036156241532, −4.57108476334353165527410767070, −4.18665102839467726616140217946, −3.04917837056806159873165981883, −2.22759715042118675508677524190, −1.07052877380793107056746654095,
0.51085983464940427718472066520, 1.98262996051101167157423933302, 2.69683237052518690835682039274, 3.77537640734545292307728557032, 4.39360323268857218136271651837, 5.51266280000727079832454480357, 5.81821549344828077650525778779, 6.91775279138683695909851030562, 7.36263954116059560524816969212, 8.528067630959460161902718560548