| L(s) = 1 | + (−0.909 − 0.415i)2-s + (0.880 − 2.99i)3-s + (0.654 + 0.755i)4-s + (−1.33 + 1.79i)5-s + (−2.04 + 2.36i)6-s + (−3.80 − 0.547i)7-s + (−0.281 − 0.959i)8-s + (−5.68 − 3.65i)9-s + (1.96 − 1.07i)10-s + (−1.34 − 2.95i)11-s + (2.84 − 1.29i)12-s + (1.88 − 0.271i)13-s + (3.23 + 2.07i)14-s + (4.19 + 5.58i)15-s + (−0.142 + 0.989i)16-s + (−4.55 − 3.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 0.293i)2-s + (0.508 − 1.73i)3-s + (0.327 + 0.377i)4-s + (−0.598 + 0.801i)5-s + (−0.835 + 0.963i)6-s + (−1.43 − 0.206i)7-s + (−0.0996 − 0.339i)8-s + (−1.89 − 1.21i)9-s + (0.620 − 0.339i)10-s + (−0.406 − 0.889i)11-s + (0.820 − 0.374i)12-s + (0.524 − 0.0753i)13-s + (0.864 + 0.555i)14-s + (1.08 + 1.44i)15-s + (−0.0355 + 0.247i)16-s + (−1.10 − 0.956i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00718726 - 0.586422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00718726 - 0.586422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 + 0.415i)T \) |
| 5 | \( 1 + (1.33 - 1.79i)T \) |
| 23 | \( 1 + (-4.49 + 1.66i)T \) |
| good | 3 | \( 1 + (-0.880 + 2.99i)T + (-2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (3.80 + 0.547i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (1.34 + 2.95i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-1.88 + 0.271i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (4.55 + 3.94i)T + (2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.42 - 2.79i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-0.494 + 0.571i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-9.29 + 2.72i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.639 + 0.995i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (3.65 - 2.35i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.624 + 2.12i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 3.45iT - 47T^{2} \) |
| 53 | \( 1 + (10.6 + 1.52i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.24 + 8.63i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.29 + 0.967i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 4.67i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.37 + 5.20i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.37 + 1.19i)T + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.926 + 6.44i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (0.361 - 0.562i)T + (-34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (2.31 + 0.680i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-0.0374 - 0.0582i)T + (-40.2 + 88.2i)T^{2} \) |
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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
−11.73152199173579152024707958438, −10.97660953712123648329604503537, −9.660114036100178776637733388229, −8.532243194386957708163332572402, −7.74946840510893427050883249961, −6.74029048716460122308184020949, −6.33256735359167041994348040376, −3.31757198130797806268548971476, −2.67570872442467966465386252039, −0.53700381520814220048652016759,
2.98133557425846405120690433787, 4.20395919521475240799641572007, 5.21501163083716019160679032162, 6.68902562834005090299009317257, 8.222667892494493357686239953089, 8.998113673008705769607556601148, 9.588129830508823892718951100776, 10.39556568165037607971699723271, 11.36737124899249209804726574674, 12.69230737443861204763140784306
