| L(s) = 1 | + (−0.809 + 0.587i)2-s + (−0.913 + 0.406i)3-s + (0.309 − 0.951i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (1.01 + 1.12i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (−0.913 − 0.406i)10-s + (4.53 − 0.964i)11-s + (0.104 + 0.994i)12-s + (0.403 − 3.83i)13-s + (−1.48 − 0.315i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (−0.709 − 0.150i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.527 + 0.234i)3-s + (0.154 − 0.475i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (0.384 + 0.426i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (−0.288 − 0.128i)10-s + (1.36 − 0.290i)11-s + (0.0301 + 0.287i)12-s + (0.111 − 1.06i)13-s + (−0.397 − 0.0843i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.171 − 0.0365i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13275 + 0.232256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13275 + 0.232256i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (1.36 + 5.39i)T \) |
| good | 7 | \( 1 + (-1.01 - 1.12i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-4.53 + 0.964i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.403 + 3.83i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (0.709 + 0.150i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.291 + 2.76i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.612 + 1.88i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.08 + 1.51i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (1.30 - 2.25i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.72 - 3.88i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.101 - 0.964i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-5.74 - 4.17i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.95 - 2.16i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-12.9 + 5.78i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + (-1.27 - 2.21i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.62 + 1.80i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-1.97 + 0.420i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-1.32 - 0.282i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.31 - 0.586i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (3.47 - 10.7i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.53 - 7.79i)T + (-78.4 - 57.0i)T^{2} \) |
| 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
−10.02350846182767845661315098634, −9.301028519855379349941393209280, −8.522026294887397990194572401550, −7.60301774547909639201284895894, −6.55901971191937082471498681503, −6.01346214765900168855668959515, −5.07811737440306765128209181384, −3.91681484992612461641814522825, −2.47703533631893410090394880030, −0.913088384437735124372796510033,
1.13668934223137445201389318106, 1.97709459024117370878937971974, 3.79750188177073020973384986747, 4.51258036306478323630476054401, 5.79076693117367225925087097762, 6.77858495455594637607820560323, 7.38960582481939994718801750214, 8.589534120416425196447715706888, 9.171234604777900702750044290356, 10.01891443423443146938961487240
