| L(s) = 1 | + (−0.157 + 2.82i)2-s + (−5.18 + 0.355i)3-s + (−7.95 − 0.888i)4-s + (−1.23 − 0.715i)5-s + (−0.188 − 14.6i)6-s + (−23.8 + 13.7i)7-s + (3.76 − 22.3i)8-s + (26.7 − 3.68i)9-s + (2.21 − 3.38i)10-s + (11.1 + 19.2i)11-s + (41.5 + 1.78i)12-s + (−34.5 + 59.9i)13-s + (−35.1 − 69.6i)14-s + (6.67 + 3.26i)15-s + (62.4 + 14.1i)16-s + 31.4i·17-s + ⋯ |
| L(s) = 1 | + (−0.0556 + 0.998i)2-s + (−0.997 + 0.0683i)3-s + (−0.993 − 0.111i)4-s + (−0.110 − 0.0639i)5-s + (−0.0127 − 0.999i)6-s + (−1.28 + 0.744i)7-s + (0.166 − 0.986i)8-s + (0.990 − 0.136i)9-s + (0.0700 − 0.107i)10-s + (0.304 + 0.527i)11-s + (0.999 + 0.0428i)12-s + (−0.738 + 1.27i)13-s + (−0.671 − 1.32i)14-s + (0.114 + 0.0562i)15-s + (0.975 + 0.220i)16-s + 0.448i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0417542 - 0.415881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0417542 - 0.415881i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.157 - 2.82i)T \) |
| 3 | \( 1 + (5.18 - 0.355i)T \) |
| good | 5 | \( 1 + (1.23 + 0.715i)T + (62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (23.8 - 13.7i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-11.1 - 19.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (34.5 - 59.9i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 31.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 11.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-72.6 + 125. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (93.6 - 54.0i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-102. - 59.3i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-344. - 199. i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (173. - 100. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-151. - 262. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 243. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-41.9 + 72.6i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-199. - 345. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-307. - 177. i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 866.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 64.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (354. - 204. i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (79.8 + 138. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.49e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-700. - 1.21e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
| 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
−16.48530988338179232768010395870, −15.77035867514978607479358886588, −14.54796471144467629046833197441, −12.86269125887893195606649411156, −12.09153507335800581171439827082, −10.08963044070322272309546182470, −9.073913246976749088333192673535, −6.99368991576023138643632344827, −6.11072861500115703918770118267, −4.47871680805328595193746332110,
0.41786789564530284073403091864, 3.53871337974721602278137700429, 5.47533694399761423387754034301, 7.32644566069192145133060134360, 9.547768912309278572779484051934, 10.45570647740804030116754396230, 11.60162663654038148290530263092, 12.79512393311917454720761703502, 13.55297885640547684695221181618, 15.52563598822469997270508225743
