L(s) = 1 | + 2-s + (0.230 − 1.71i)3-s + 4-s + (0.286 + 2.21i)5-s + (0.230 − 1.71i)6-s + 4.09i·7-s + 8-s + (−2.89 − 0.791i)9-s + (0.286 + 2.21i)10-s + 2.91·11-s + (0.230 − 1.71i)12-s − 2.84·13-s + 4.09i·14-s + (3.87 + 0.0193i)15-s + 16-s + 3.40i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.133 − 0.991i)3-s + 0.5·4-s + (0.128 + 0.991i)5-s + (0.0941 − 0.700i)6-s + 1.54i·7-s + 0.353·8-s + (−0.964 − 0.263i)9-s + (0.0906 + 0.701i)10-s + 0.879·11-s + (0.0665 − 0.495i)12-s − 0.790·13-s + 1.09i·14-s + (0.999 + 0.00499i)15-s + 0.250·16-s + 0.826i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39027 + 0.758387i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39027 + 0.758387i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.230 + 1.71i)T \) |
| 5 | \( 1 + (-0.286 - 2.21i)T \) |
| 31 | \( 1 + (-4.53 + 3.23i)T \) |
good | 7 | \( 1 - 4.09iT - 7T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 - 3.40iT - 17T^{2} \) |
| 19 | \( 1 - 2.22T + 19T^{2} \) |
| 23 | \( 1 - 3.74iT - 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 + 0.438iT - 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 - 7.81T + 47T^{2} \) |
| 53 | \( 1 + 10.9iT - 53T^{2} \) |
| 59 | \( 1 + 2.26iT - 59T^{2} \) |
| 61 | \( 1 + 3.00iT - 61T^{2} \) |
| 67 | \( 1 + 9.68iT - 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 2.62T + 73T^{2} \) |
| 79 | \( 1 + 15.1iT - 79T^{2} \) |
| 83 | \( 1 + 8.73iT - 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 3.32iT - 97T^{2} \) |
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
−10.18650918070382752490846742326, −9.264170239545182748024855873464, −8.305967152069283994589542802841, −7.43884707739450856561592295054, −6.48051477744030151345349667043, −6.07397958431553852388427991863, −5.11475951479931199232355072474, −3.51989470535819526732125195763, −2.64389349376891265960021465905, −1.83383424339267806460981834561,
0.971422897977406397862759727089, 2.81767411373751769999056228503, 4.02075950646901070439227670031, 4.53042346316606603550378718231, 5.21542825018905061862093651073, 6.45658690748875815572480772264, 7.38555845123359274463032532386, 8.374065478044939669663202457226, 9.335571819045600490318814124033, 10.04259537380149329471533031594