| L(s) = 1 | + (−1.38 − 0.296i)2-s + (1.89 + 0.661i)3-s + (1.82 + 0.820i)4-s + (−0.323 − 0.0737i)5-s + (−2.41 − 1.47i)6-s + (1.02 + 2.12i)7-s + (−2.27 − 1.67i)8-s + (0.789 + 0.629i)9-s + (0.424 + 0.197i)10-s + (−0.0717 + 0.636i)11-s + (2.90 + 2.75i)12-s + (0.943 − 0.752i)13-s + (−0.782 − 3.23i)14-s + (−0.562 − 0.353i)15-s + (2.65 + 2.99i)16-s + (4.68 − 4.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.977 − 0.209i)2-s + (1.09 + 0.381i)3-s + (0.911 + 0.410i)4-s + (−0.144 − 0.0329i)5-s + (−0.986 − 0.602i)6-s + (0.386 + 0.801i)7-s + (−0.805 − 0.592i)8-s + (0.263 + 0.209i)9-s + (0.134 + 0.0625i)10-s + (−0.0216 + 0.192i)11-s + (0.838 + 0.796i)12-s + (0.261 − 0.208i)13-s + (−0.209 − 0.865i)14-s + (−0.145 − 0.0911i)15-s + (0.663 + 0.748i)16-s + (1.13 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.944894 + 0.137678i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.944894 + 0.137678i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.296i)T \) |
| 29 | \( 1 + (5.17 + 1.47i)T \) |
| good | 3 | \( 1 + (-1.89 - 0.661i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (0.323 + 0.0737i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (-1.02 - 2.12i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (0.0717 - 0.636i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (-0.943 + 0.752i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-4.68 + 4.68i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.361 - 1.03i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (7.56 - 1.72i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-1.27 + 0.800i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (5.44 - 0.613i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-7.01 - 7.01i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.87 + 4.57i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (4.19 + 0.472i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (-0.531 + 2.32i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 6.44iT - 59T^{2} \) |
| 61 | \( 1 + (2.72 - 7.77i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (6.60 - 8.28i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-5.25 - 6.59i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-6.38 + 10.1i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 1.20i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (7.30 - 15.1i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-1.07 - 1.70i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (-1.40 - 4.01i)T + (-75.8 + 60.4i)T^{2} \) |
| 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
−13.80372094313992096368400584801, −12.21222389485104126359500178047, −11.54439403471371290779056979356, −10.00234773539933768325201989726, −9.377605608006612782363739795488, −8.288061089807089505884323026460, −7.65523141557830367442011439717, −5.80907734296418799738425840636, −3.63219194945237549901039546119, −2.28342878669183143509246284506,
1.80299074239515416920515668683, 3.61729093180464051151534763288, 5.94339850321471907099264412713, 7.51616286868857069454340721189, 7.970110293000209839081055512379, 9.022680178420227985124960228982, 10.19047040599889950458914020585, 11.14924927722533875700632066026, 12.45243454116555485759812680363, 13.88950468745651095529072682268
