| L(s) = 1 | + (0.104 + 0.994i)2-s + (−1.36 − 1.52i)3-s + (−0.978 + 0.207i)4-s + (2.23 − 0.100i)5-s + (1.36 − 1.52i)6-s + 5.15·7-s + (−0.309 − 0.951i)8-s + (−0.124 + 1.18i)9-s + (0.333 + 2.21i)10-s + (1.20 − 0.878i)11-s + (1.65 + 1.20i)12-s + (−0.0653 + 0.622i)13-s + (0.538 + 5.12i)14-s + (−3.21 − 3.25i)15-s + (0.913 − 0.406i)16-s + (3.02 + 0.642i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.790 − 0.878i)3-s + (−0.489 + 0.103i)4-s + (0.998 − 0.0451i)5-s + (0.559 − 0.621i)6-s + 1.94·7-s + (−0.109 − 0.336i)8-s + (−0.0414 + 0.394i)9-s + (0.105 + 0.699i)10-s + (0.364 − 0.264i)11-s + (0.478 + 0.347i)12-s + (−0.0181 + 0.172i)13-s + (0.144 + 1.37i)14-s + (−0.829 − 0.841i)15-s + (0.228 − 0.101i)16-s + (0.732 + 0.155i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80376 - 0.0822715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80376 - 0.0822715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-2.23 + 0.100i)T \) |
| 19 | \( 1 + (3.76 - 2.19i)T \) |
| good | 3 | \( 1 + (1.36 + 1.52i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 - 5.15T + 7T^{2} \) |
| 11 | \( 1 + (-1.20 + 0.878i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.0653 - 0.622i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-3.02 - 0.642i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (0.685 + 0.305i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-0.434 + 0.0923i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (0.565 + 1.74i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.25 + 2.36i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.38 + 0.616i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.87 + 3.24i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (7.09 - 1.50i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-3.98 + 0.846i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-8.10 + 3.61i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-3.31 - 1.47i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (5.50 - 6.11i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (2.29 + 2.54i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.896 - 8.53i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-7.43 - 8.26i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (3.27 + 10.0i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-12.6 - 5.61i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (9.12 + 10.1i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.07530639406868736756659334045, −8.900171762273652156505645312817, −8.222833579993282404601951951930, −7.38053037432246072328298817018, −6.53562552070291956024954251863, −5.70404902929609133031837712624, −5.21677533011643331716100578599, −4.08580990602001305678450128906, −2.02545064176860506720119707318, −1.15778227988937148679188922446,
1.35358288571639548343950594825, 2.35036880329983945226607157351, 3.99425893064356540388536816703, 4.99685791534770513866343662348, 5.17158589345550696083397849567, 6.30949609926785446593450244961, 7.69720341058729720678829273815, 8.638159899981267790657433672729, 9.469027607412727359906068874776, 10.42505409819833772391522228863
