L(s) = 1 | + (0.727 − 0.528i)2-s + (−0.367 + 1.13i)4-s + (−0.650 − 0.472i)5-s + (0.309 − 0.951i)7-s + (0.887 + 2.73i)8-s − 0.723·10-s + (3.31 + 0.199i)11-s + (−2.38 + 1.73i)13-s + (−0.278 − 0.855i)14-s + (0.163 + 0.119i)16-s + (2.67 + 1.94i)17-s + (2.45 + 7.55i)19-s + (0.774 − 0.562i)20-s + (2.51 − 1.60i)22-s + 7.53·23-s + ⋯ |
L(s) = 1 | + (0.514 − 0.373i)2-s + (−0.183 + 0.566i)4-s + (−0.290 − 0.211i)5-s + (0.116 − 0.359i)7-s + (0.313 + 0.965i)8-s − 0.228·10-s + (0.998 + 0.0602i)11-s + (−0.661 + 0.480i)13-s + (−0.0743 − 0.228i)14-s + (0.0409 + 0.0297i)16-s + (0.648 + 0.471i)17-s + (0.562 + 1.73i)19-s + (0.173 − 0.125i)20-s + (0.536 − 0.342i)22-s + 1.57·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78602 + 0.498386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78602 + 0.498386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.31 - 0.199i)T \) |
good | 2 | \( 1 + (-0.727 + 0.528i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (0.650 + 0.472i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2.38 - 1.73i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.67 - 1.94i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.45 - 7.55i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 7.53T + 23T^{2} \) |
| 29 | \( 1 + (-0.0724 + 0.222i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.22 - 2.34i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.165 - 0.510i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.47 - 4.55i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 + (-3.40 - 10.4i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.49 + 6.89i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.62 + 4.99i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.95 + 2.14i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + (8.36 + 6.07i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.75 + 8.47i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.77 + 4.92i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.87 + 4.26i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 8.03T + 89T^{2} \) |
| 97 | \( 1 + (15.5 - 11.3i)T + (29.9 - 92.2i)T^{2} \) |
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
−10.70368745567849841276980274491, −9.677303971597249141238021452568, −8.795128691527715106411180225340, −7.88301671410741975743037455931, −7.19665025316318175919375091063, −5.93064218952551963115125999336, −4.75523258528294778448651515156, −3.99984453197988356992405666854, −3.13267789185970017695042853793, −1.55577959595657241991024621669,
0.958272979930099444101306062250, 2.80418003890921067950380306389, 4.01935965753391422523422431833, 5.11234475406836017977863477493, 5.66510325426693209725910006211, 7.05986374624694207515096728701, 7.25263471520008240074775219751, 8.972738131079505653274846114432, 9.323749963960820304756922451709, 10.41245754010964280867666883026