| L(s) = 1 | + (0.959 − 0.281i)2-s + (−0.267 − 0.308i)3-s + (0.841 − 0.540i)4-s + (0.142 + 0.989i)5-s + (−0.343 − 0.220i)6-s + (1.05 − 2.30i)7-s + (0.654 − 0.755i)8-s + (0.403 − 2.80i)9-s + (0.415 + 0.909i)10-s + (0.989 + 0.290i)11-s + (−0.391 − 0.115i)12-s + (1.97 + 4.31i)13-s + (0.360 − 2.50i)14-s + (0.267 − 0.308i)15-s + (0.415 − 0.909i)16-s + (2.04 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (−0.154 − 0.178i)3-s + (0.420 − 0.270i)4-s + (0.0636 + 0.442i)5-s + (−0.140 − 0.0900i)6-s + (0.397 − 0.870i)7-s + (0.231 − 0.267i)8-s + (0.134 − 0.934i)9-s + (0.131 + 0.287i)10-s + (0.298 + 0.0875i)11-s + (−0.113 − 0.0331i)12-s + (0.547 + 1.19i)13-s + (0.0962 − 0.669i)14-s + (0.0690 − 0.0796i)15-s + (0.103 − 0.227i)16-s + (0.495 + 0.318i)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
| \(L(1)\) |
\(\approx\) |
\(1.70191 - 0.539790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70191 - 0.539790i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.0958 + 4.79i)T \) |
| good | 3 | \( 1 + (0.267 + 0.308i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.05 + 2.30i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.989 - 0.290i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.97 - 4.31i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.04 - 1.31i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (5.54 - 3.56i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (8.51 + 5.46i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.98 - 2.29i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.65 - 11.4i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.933 - 6.49i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (0.689 + 0.795i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 + (3.57 - 7.81i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.89 + 10.7i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (1.43 - 1.66i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-9.47 + 2.78i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.24 + 0.951i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-12.4 + 7.97i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.79 + 3.93i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.515 + 3.58i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (3.82 + 4.40i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.252 - 1.75i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.12031762455399911173872245054, −11.26183162949065457261782657554, −10.45284268655422575774165124884, −9.371726486064844544371621464173, −7.969002157619853268293672405897, −6.69354093759713232494344732754, −6.18731421834301262598088713678, −4.41159436383242542001803105382, −3.63263938099214256587689391844, −1.66353495445311367768795878789,
2.15828278116003507446149490017, 3.81913269579198233524150789930, 5.28922348815654301619487912401, 5.62716604792888386202577433029, 7.28561474045145439477511419094, 8.292065688881597075386329851739, 9.247493122679936208148714102691, 10.73553905772378327280655868280, 11.31259809730137386232394433397, 12.56691550431795066074860170901
