L(s) = 1 | + (−2.44 + 2.44i)5-s + 1.41i·7-s + (3.46 − 3.46i)11-s + (1 + i)13-s + 4.89·17-s + (4.24 + 4.24i)19-s − 6.92i·23-s − 6.99i·25-s + (−2.44 − 2.44i)29-s + 1.41·31-s + (−3.46 − 3.46i)35-s + (−7 + 7i)37-s − 4.89i·41-s + (−4.24 + 4.24i)43-s + 6.92·47-s + ⋯ |
L(s) = 1 | + (−1.09 + 1.09i)5-s + 0.534i·7-s + (1.04 − 1.04i)11-s + (0.277 + 0.277i)13-s + 1.18·17-s + (0.973 + 0.973i)19-s − 1.44i·23-s − 1.39i·25-s + (−0.454 − 0.454i)29-s + 0.254·31-s + (−0.585 − 0.585i)35-s + (−1.15 + 1.15i)37-s − 0.765i·41-s + (−0.646 + 0.646i)43-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.556071392\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.556071392\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.44 - 2.44i)T - 5iT^{2} \) |
| 7 | \( 1 - 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (-3.46 + 3.46i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + (-4.24 - 4.24i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + (2.44 + 2.44i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + (7 - 7i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 + (4.24 - 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + (-2.44 + 2.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.92 - 6.92i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5 - 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.65 - 5.65i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + (-10.3 - 10.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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
−8.992521338353650964187968394745, −8.312579251204752996531685439662, −7.66851200892466053335876738600, −6.79123032816059840875573520111, −6.17360557421327555411935147917, −5.31616736129547997428477704072, −3.91608617923241368305311004346, −3.54101889266067087789320907904, −2.62749553011673268060178648233, −1.04435732686169025551394273290,
0.71011411293060810881502761008, 1.61411652970955365868324454332, 3.45149555960300229446798627646, 3.87750669352539003630034460054, 4.88885521163887910695988516416, 5.42509691546849037865222465397, 6.77310305251654785431750551022, 7.52465618502497887511208441763, 7.86480942624986109808327009713, 9.079759713950365411539848365268