L(s) = 1 | + (0.291 + 2.98i)3-s + 4.46·7-s + (−8.82 + 1.74i)9-s − 17.8i·11-s + 11.0·13-s + 0.794i·17-s − 26.5·19-s + (1.30 + 13.3i)21-s − 14.9i·23-s + (−7.77 − 25.8i)27-s − 5.58i·29-s − 53.1·31-s + (53.3 − 5.21i)33-s − 51.7·37-s + (3.21 + 32.8i)39-s + ⋯ |
L(s) = 1 | + (0.0972 + 0.995i)3-s + 0.637·7-s + (−0.981 + 0.193i)9-s − 1.62i·11-s + 0.846·13-s + 0.0467i·17-s − 1.39·19-s + (0.0619 + 0.634i)21-s − 0.649i·23-s + (−0.287 − 0.957i)27-s − 0.192i·29-s − 1.71·31-s + (1.61 − 0.157i)33-s − 1.39·37-s + (0.0823 + 0.842i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0972 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0972 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.034720878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.034720878\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.291 - 2.98i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.46T + 49T^{2} \) |
| 11 | \( 1 + 17.8iT - 121T^{2} \) |
| 13 | \( 1 - 11.0T + 169T^{2} \) |
| 17 | \( 1 - 0.794iT - 289T^{2} \) |
| 19 | \( 1 + 26.5T + 361T^{2} \) |
| 23 | \( 1 + 14.9iT - 529T^{2} \) |
| 29 | \( 1 + 5.58iT - 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 + 51.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 67.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 12.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 37.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 97.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 3.02T + 4.48e3T^{2} \) |
| 71 | \( 1 + 57.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 2.16T + 6.24e3T^{2} \) |
| 83 | \( 1 - 13.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 173. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 91.6T + 9.40e3T^{2} \) |
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
−9.182982939259388406161571649848, −8.533061233690061934271336880633, −8.205429766097006023379488569302, −6.72240439189961789962466152516, −5.80280169424138795366198717116, −5.13499873572905998888790188500, −3.97143373521287650025092893608, −3.39906540223322574100606659914, −2.02156572510541823525350677449, −0.28595368234134848833708724313,
1.51578314056836495300853496257, 2.09124442245925906429052609781, 3.53174872664939624530941930878, 4.66265062206701386572841883434, 5.59885648973217779999940773955, 6.65795829042883989865910533061, 7.20449843296804396913667938874, 8.105938775740010596724402941720, 8.713955850950206785270143056888, 9.664644405438894846596173236197