| L(s) = 1 | + (0.281 − 0.959i)2-s + (−2.36 − 2.05i)3-s + (−0.841 − 0.540i)4-s + (−0.736 − 2.11i)5-s + (−2.63 + 1.69i)6-s + (1.55 − 0.710i)7-s + (−0.755 + 0.654i)8-s + (0.972 + 6.76i)9-s + (−2.23 + 0.111i)10-s + (−0.297 + 0.0874i)11-s + (0.883 + 3.00i)12-s + (−2.61 − 1.19i)13-s + (−0.243 − 1.69i)14-s + (−2.59 + 6.51i)15-s + (0.415 + 0.909i)16-s + (2.98 + 4.65i)17-s + ⋯ |
| L(s) = 1 | + (0.199 − 0.678i)2-s + (−1.36 − 1.18i)3-s + (−0.420 − 0.270i)4-s + (−0.329 − 0.944i)5-s + (−1.07 + 0.692i)6-s + (0.588 − 0.268i)7-s + (−0.267 + 0.231i)8-s + (0.324 + 2.25i)9-s + (−0.706 + 0.0353i)10-s + (−0.0897 + 0.0263i)11-s + (0.254 + 0.868i)12-s + (−0.723 − 0.330i)13-s + (−0.0650 − 0.452i)14-s + (−0.668 + 1.68i)15-s + (0.103 + 0.227i)16-s + (0.725 + 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.182002 + 0.573797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182002 + 0.573797i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.281 + 0.959i)T \) |
| 5 | \( 1 + (0.736 + 2.11i)T \) |
| 23 | \( 1 + (-0.555 + 4.76i)T \) |
| good | 3 | \( 1 + (2.36 + 2.05i)T + (0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.55 + 0.710i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.297 - 0.0874i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.61 + 1.19i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.98 - 4.65i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.72 + 2.39i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-7.45 + 4.79i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (5.47 + 6.31i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (5.64 - 0.811i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.124 - 0.869i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.72 - 3.22i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 + (3.19 - 1.46i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.22 + 4.86i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (3.37 + 3.89i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (0.478 - 1.62i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.87 - 2.31i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.64 + 2.55i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.42 - 7.49i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (0.701 - 0.100i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-6.06 + 6.99i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-17.0 - 2.45i)T + (93.0 + 27.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.79276212568482833615341215468, −11.00367168050356564201627976281, −10.12893675341284680415432905759, −8.440723360245643291378948168706, −7.66272650573547001264882867482, −6.32075571731166756457525633028, −5.25181279981816828601771696241, −4.41258462880237015848891044808, −1.94288924674232773394238346804, −0.55531010188257528611733562465,
3.42573374823449669770404492061, 4.70983329457910172795019821470, 5.43460662378505129760880283050, 6.55885244976580582207403614232, 7.49692280946689506606322542799, 9.032865238575484471497694897749, 10.10765934082735829796931131064, 10.82633362128303952997456823378, 11.77635310846618195900619053738, 12.34280655812973554981591530012
