| L(s) = 1 | + (0.415 − 0.909i)2-s + (1.63 + 0.479i)3-s + (−0.654 − 0.755i)4-s + (2.49 − 1.60i)5-s + (1.11 − 1.28i)6-s + (−0.142 + 0.989i)7-s + (−0.959 + 0.281i)8-s + (−0.0907 − 0.0583i)9-s + (−0.421 − 2.93i)10-s + (−1.24 − 2.73i)11-s + (−0.706 − 1.54i)12-s + (0.366 + 2.54i)13-s + (0.841 + 0.540i)14-s + (4.83 − 1.41i)15-s + (−0.142 + 0.989i)16-s + (−2.46 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (0.942 + 0.276i)3-s + (−0.327 − 0.377i)4-s + (1.11 − 0.716i)5-s + (0.454 − 0.524i)6-s + (−0.0537 + 0.374i)7-s + (−0.339 + 0.0996i)8-s + (−0.0302 − 0.0194i)9-s + (−0.133 − 0.927i)10-s + (−0.376 − 0.823i)11-s + (−0.203 − 0.446i)12-s + (0.101 + 0.706i)13-s + (0.224 + 0.144i)14-s + (1.24 − 0.366i)15-s + (−0.0355 + 0.247i)16-s + (−0.598 + 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.90204 - 0.992969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90204 - 0.992969i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.64 + 4.00i)T \) |
| good | 3 | \( 1 + (-1.63 - 0.479i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (-2.49 + 1.60i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (1.24 + 2.73i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.366 - 2.54i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.46 - 2.84i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.20 - 4.85i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (3.15 - 3.64i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-6.55 + 1.92i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (2.21 + 1.42i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (7.35 - 4.72i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-4.52 - 1.32i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 6.00T + 47T^{2} \) |
| 53 | \( 1 + (-0.0521 + 0.362i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.30 + 9.10i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (7.71 - 2.26i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.20 - 7.00i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.38 - 7.40i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.33 - 1.54i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.35 - 9.39i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (10.0 + 6.45i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-13.7 - 4.02i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-1.59 + 1.02i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.56762334267180618987145615985, −10.35894712145654485886378904442, −9.568891850944542554079129615984, −8.848053910288968236199330265296, −8.195603090614612899971993540703, −6.22702234300832879477852997558, −5.46635861619874958791113830024, −4.11158729777907898547050445237, −2.89014090579786832431198557605, −1.72734934847807007761856127363,
2.27326968717029574541861916287, 3.20949033524301398389833294696, 4.89739985191791061461765616340, 5.96188250583138897158722029950, 7.14262485222338738212435838932, 7.66470454407146987238010997978, 8.943390051340829097985062772813, 9.720912371351971140785420332003, 10.61997854499375795659947567826, 11.91439879343205675666218350728
