L(s) = 1 | + (1 + 1.41i)3-s − 2·5-s + (3.94 + 2.28i)7-s + (−1.00 + 2.82i)9-s + (−1.72 + 2.98i)11-s + (4.5 − 2.59i)13-s + (−2 − 2.82i)15-s + (2.17 − 1.25i)17-s + (−1.5 − 2.59i)19-s + (0.724 + 7.86i)21-s + (−8.17 + 4.71i)23-s − 25-s + (−5.00 + 1.41i)27-s + (7.62 + 4.40i)29-s + (−1.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (0.577 + 0.816i)3-s − 0.894·5-s + (1.49 + 0.861i)7-s + (−0.333 + 0.942i)9-s + (−0.520 + 0.900i)11-s + (1.24 − 0.720i)13-s + (−0.516 − 0.730i)15-s + (0.527 − 0.304i)17-s + (−0.344 − 0.596i)19-s + (0.158 + 1.71i)21-s + (−1.70 + 0.984i)23-s − 0.200·25-s + (−0.962 + 0.272i)27-s + (1.41 + 0.817i)29-s + (−0.269 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11660 + 1.35919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11660 + 1.35919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 67 | \( 1 + (-8 - 1.73i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + (-3.94 - 2.28i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.72 - 2.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.17 + 1.25i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (8.17 - 4.71i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.62 - 4.40i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0505 + 0.0874i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.724 + 1.25i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 12.5iT - 43T^{2} \) |
| 47 | \( 1 + (2.72 + 1.57i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6.29iT - 59T^{2} \) |
| 61 | \( 1 + (0.398 - 0.230i)T + (30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-3.82 - 2.20i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.94 + 3.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.39 + 5.42i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.07 + 4.08i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 + (2.84 - 1.64i)T + (48.5 - 84.0i)T^{2} \) |
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
−10.51813824591413610796329953085, −9.652637625143854450468510491701, −8.520992176267362542142757040823, −8.146149268498349776926868601450, −7.52704863177309229765839636218, −5.81394908606086645678508820606, −4.95319984068026430352370400162, −4.21355485292172450076571675996, −3.10452491063081726081917585052, −1.85586136177714784575576231631,
0.874407742149730763175277766597, 2.08872438393129253203191730676, 3.72743960242981800705260923880, 4.19633567698243252583469471942, 5.74917165277786606635523765439, 6.69619877314121659852713035107, 7.80691328418299928860344364564, 8.213904348530514521097646089770, 8.589074636065075732005220362414, 10.22285723760322320130437277464