| L(s) = 1 | + (−1.39 + 0.221i)2-s + (0.786 − 1.54i)3-s + (1.90 − 0.618i)4-s + (−4.97 + 0.522i)5-s + (−0.756 + 2.32i)6-s + (3.51 − 3.51i)7-s + (−2.52 + 1.28i)8-s + (−1.76 − 2.42i)9-s + (6.83 − 1.82i)10-s + (−11.9 − 8.70i)11-s + (0.541 − 3.42i)12-s + (−3.70 − 0.586i)13-s + (−4.13 + 5.68i)14-s + (−3.10 + 8.08i)15-s + (3.23 − 2.35i)16-s + (1.67 + 3.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.110i)2-s + (0.262 − 0.514i)3-s + (0.475 − 0.154i)4-s + (−0.994 + 0.104i)5-s + (−0.126 + 0.388i)6-s + (0.502 − 0.502i)7-s + (−0.315 + 0.160i)8-s + (−0.195 − 0.269i)9-s + (0.683 − 0.182i)10-s + (−1.08 − 0.791i)11-s + (0.0451 − 0.285i)12-s + (−0.284 − 0.0451i)13-s + (−0.295 + 0.406i)14-s + (−0.206 + 0.538i)15-s + (0.202 − 0.146i)16-s + (0.0983 + 0.193i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.189085 - 0.508569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189085 - 0.508569i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.221i)T \) |
| 3 | \( 1 + (-0.786 + 1.54i)T \) |
| 5 | \( 1 + (4.97 - 0.522i)T \) |
| good | 7 | \( 1 + (-3.51 + 3.51i)T - 49iT^{2} \) |
| 11 | \( 1 + (11.9 + 8.70i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (3.70 + 0.586i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-1.67 - 3.28i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (31.9 + 10.3i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (3.40 + 21.5i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (32.1 - 10.4i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-14.5 + 44.9i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (0.945 - 5.97i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-27.4 + 19.9i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-55.9 - 55.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-49.3 - 25.1i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-3.34 + 6.56i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-34.2 - 47.1i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (17.6 + 12.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (4.02 + 7.90i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (30.3 + 93.4i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (14.8 + 94.0i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (13.2 - 4.31i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-94.6 + 48.2i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (6.26 - 8.61i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (43.2 + 22.0i)T + (5.53e3 + 7.61e3i)T^{2} \) |
| 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
−12.36449459560559635255827300849, −11.05363109671509196451475174613, −10.67648874767291866807893675486, −8.972507178893740423092686397253, −8.008062409791836865459295032646, −7.50356568527077655475249170706, −6.14161912833287284412509741819, −4.33220563219361763780837899347, −2.58225204677600752294139046023, −0.40205517294268510186259174225,
2.33475188312920402641776345753, 4.02858396069815067260570668832, 5.34482814468502384996908996101, 7.23978084896934287528039023616, 8.100580021561512635667824326651, 8.927041002892712755384838591339, 10.17391528625016224094674021339, 10.97864766282440330939656916272, 12.02773653052371375691574979422, 12.85413258445370951483720845932
