| L(s) = 1 | + (2.68 − 0.872i)2-s + (0.0862 + 0.0626i)3-s + (3.21 − 2.33i)4-s + (−0.690 + 2.12i)5-s + (0.286 + 0.0929i)6-s + (−1.76 − 2.42i)7-s + (−0.0523 + 0.0721i)8-s + (−2.77 − 8.54i)9-s + 6.31i·10-s + (−4.57 + 10.0i)11-s + 0.423·12-s + (0.925 − 0.300i)13-s + (−6.85 − 4.98i)14-s + (−0.192 + 0.140i)15-s + (−4.98 + 15.3i)16-s + (4.85 + 1.57i)17-s + ⋯ |
| L(s) = 1 | + (1.34 − 0.436i)2-s + (0.0287 + 0.0208i)3-s + (0.802 − 0.583i)4-s + (−0.138 + 0.425i)5-s + (0.0476 + 0.0154i)6-s + (−0.252 − 0.347i)7-s + (−0.00654 + 0.00901i)8-s + (−0.308 − 0.949i)9-s + 0.631i·10-s + (−0.416 + 0.909i)11-s + 0.0352·12-s + (0.0711 − 0.0231i)13-s + (−0.489 − 0.355i)14-s + (−0.0128 + 0.00934i)15-s + (−0.311 + 0.958i)16-s + (0.285 + 0.0928i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.94267 - 0.401101i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94267 - 0.401101i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.690 - 2.12i)T \) |
| 11 | \( 1 + (4.57 - 10.0i)T \) |
| good | 2 | \( 1 + (-2.68 + 0.872i)T + (3.23 - 2.35i)T^{2} \) |
| 3 | \( 1 + (-0.0862 - 0.0626i)T + (2.78 + 8.55i)T^{2} \) |
| 7 | \( 1 + (1.76 + 2.42i)T + (-15.1 + 46.6i)T^{2} \) |
| 13 | \( 1 + (-0.925 + 0.300i)T + (136. - 99.3i)T^{2} \) |
| 17 | \( 1 + (-4.85 - 1.57i)T + (233. + 169. i)T^{2} \) |
| 19 | \( 1 + (-7.85 + 10.8i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 - 24.5T + 529T^{2} \) |
| 29 | \( 1 + (19.2 + 26.5i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (3.40 + 10.4i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (7.65 - 5.56i)T + (423. - 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-43.5 + 59.9i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 - 52.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.4 - 24.2i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-19.3 - 59.4i)T + (-2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (57.2 - 41.5i)T + (1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (69.3 + 22.5i)T + (3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + 116.T + 4.48e3T^{2} \) |
| 71 | \( 1 + (38.9 - 119. i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (11.6 + 16.0i)T + (-1.64e3 + 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-31.0 + 10.1i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-138. - 44.9i)T + (5.57e3 + 4.04e3i)T^{2} \) |
| 89 | \( 1 + 94.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (42.4 + 130. i)T + (-7.61e3 + 5.53e3i)T^{2} \) |
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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
−14.88508264713138928928529879345, −13.77968629157741768673247330860, −12.77587425086690289790081635571, −11.86118928720529937392493138104, −10.74395106275120031756254568431, −9.263255174924732416231533400933, −7.27933255900320559353014299325, −5.85697510858194832709784940879, −4.27501513238284849703587537622, −2.91575294438019046833768318308,
3.20519655280078667862854042591, 4.97712269824737475144473752257, 5.88650060530106413442106181285, 7.57144632210813432911226398510, 9.025464779849471229055407762960, 10.83722752289923113190338068657, 12.13579305330249568051728984930, 13.15418370192447636378238968768, 13.89814204294550359394449360438, 14.98092410464650716682704554958
