| 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
−11.91439879343205675666218350728, −10.61997854499375795659947567826, −9.720912371351971140785420332003, −8.943390051340829097985062772813, −7.66470454407146987238010997978, −7.14262485222338738212435838932, −5.96188250583138897158722029950, −4.89739985191791061461765616340, −3.20949033524301398389833294696, −2.27326968717029574541861916287,
1.72734934847807007761856127363, 2.89014090579786832431198557605, 4.11158729777907898547050445237, 5.46635861619874958791113830024, 6.22702234300832879477852997558, 8.195603090614612899971993540703, 8.848053910288968236199330265296, 9.568891850944542554079129615984, 10.35894712145654485886378904442, 11.56762334267180618987145615985
