L(s) = 1 | − i·2-s + (−1.22 − 1.22i)3-s − 4-s + i·5-s + (−1.22 + 1.22i)6-s − 4·7-s + i·8-s + 2.99i·9-s + 10-s − 4.89·11-s + (1.22 + 1.22i)12-s − 2.44i·13-s + 4i·14-s + (1.22 − 1.22i)15-s + 16-s + 7.34·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.707 − 0.707i)3-s − 0.5·4-s + 0.447i·5-s + (−0.499 + 0.499i)6-s − 1.51·7-s + 0.353i·8-s + 0.999i·9-s + 0.316·10-s − 1.47·11-s + (0.353 + 0.353i)12-s − 0.679i·13-s + 1.06i·14-s + (0.316 − 0.316i)15-s + 0.250·16-s + 1.78·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.677744 - 0.112806i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.677744 - 0.112806i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 - iT \) |
| 31 | \( 1 + (-5 + 2.44i)T \) |
good | 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7.34T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 - 2.44iT - 37T^{2} \) |
| 41 | \( 1 - 12iT - 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 2.44iT - 73T^{2} \) |
| 79 | \( 1 + 4.89iT - 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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
−10.18249778163730556648966859896, −9.615763821407096304551392312626, −8.147018753252045796441529653789, −7.55555001524775968721659651400, −6.47700255850120500820894218825, −5.74268044766318991151298087838, −4.85181410486138855101146471851, −3.13479123036722053212503588555, −2.75901749428737794613219065637, −0.892895951309314894610010250955,
0.49408779239236033858723886682, 3.01699969319885962967433074306, 3.95417809015236702877811425878, 5.19668567932328265085813598065, 5.58541357955660783925553677735, 6.63484674152703722280720964158, 7.35206537591218777684528712626, 8.600750132073482232153268457300, 9.295299000125931295921043648823, 10.16402540634935931500742880294