| L(s) = 1 | + (−0.180 − 2.00i)2-s + (−2.01 + 0.365i)4-s + (0.521 − 0.717i)5-s + (−0.352 − 0.590i)7-s + (0.0251 + 0.0910i)8-s + (−1.53 − 0.914i)10-s + (−0.768 + 1.16i)11-s + (−4.65 + 3.07i)13-s + (−1.11 + 0.813i)14-s + (−3.65 + 1.37i)16-s + (2.27 − 6.99i)17-s + (−5.61 − 5.87i)19-s + (−0.787 + 1.63i)20-s + (2.47 + 1.32i)22-s + (−1.32 − 5.80i)23-s + ⋯ |
| L(s) = 1 | + (−0.127 − 1.41i)2-s + (−1.00 + 0.182i)4-s + (0.233 − 0.320i)5-s + (−0.133 − 0.223i)7-s + (0.00888 + 0.0322i)8-s + (−0.484 − 0.289i)10-s + (−0.231 + 0.350i)11-s + (−1.29 + 0.851i)13-s + (−0.299 + 0.217i)14-s + (−0.913 + 0.342i)16-s + (0.551 − 1.69i)17-s + (−1.28 − 1.34i)19-s + (−0.176 + 0.365i)20-s + (0.526 + 0.283i)22-s + (−0.276 − 1.20i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.272458 + 0.808505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272458 + 0.808505i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (-6.74 + 5.05i)T \) |
| good | 2 | \( 1 + (0.180 + 2.00i)T + (-1.96 + 0.357i)T^{2} \) |
| 5 | \( 1 + (-0.521 + 0.717i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.352 + 0.590i)T + (-3.31 + 6.16i)T^{2} \) |
| 11 | \( 1 + (0.768 - 1.16i)T + (-4.32 - 10.1i)T^{2} \) |
| 13 | \( 1 + (4.65 - 3.07i)T + (5.10 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-2.27 + 6.99i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.61 + 5.87i)T + (-0.852 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.32 + 5.80i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (1.96 - 1.05i)T + (15.9 - 24.2i)T^{2} \) |
| 31 | \( 1 + (-2.21 + 5.88i)T + (-23.3 - 20.3i)T^{2} \) |
| 37 | \( 1 + (1.56 - 6.84i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-1.13 + 1.42i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-1.99 + 1.74i)T + (5.77 - 42.6i)T^{2} \) |
| 47 | \( 1 + (-0.987 - 7.29i)T + (-45.3 + 12.5i)T^{2} \) |
| 53 | \( 1 + (6.83 + 1.24i)T + (49.6 + 18.6i)T^{2} \) |
| 59 | \( 1 + (0.111 + 2.49i)T + (-58.7 + 5.28i)T^{2} \) |
| 61 | \( 1 + (0.319 - 0.535i)T + (-28.9 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-2.86 - 15.7i)T + (-62.7 + 23.5i)T^{2} \) |
| 73 | \( 1 + (1.50 - 0.135i)T + (71.8 - 13.0i)T^{2} \) |
| 79 | \( 1 + (-10.3 + 2.84i)T + (67.8 - 40.5i)T^{2} \) |
| 83 | \( 1 + (-6.14 + 0.276i)T + (82.6 - 7.44i)T^{2} \) |
| 89 | \( 1 + (-1.54 + 8.51i)T + (-83.3 - 31.2i)T^{2} \) |
| 97 | \( 1 + (-11.1 + 8.87i)T + (21.5 - 94.5i)T^{2} \) |
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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
−10.01157028976697266899889636795, −9.505613030841015976472874406932, −8.790068343930345456150600603482, −7.35335627452226424495242350972, −6.61824294942208819662255412513, −4.93930750335356916445797719942, −4.37455799242821789165253582060, −2.85424572058681982083628659016, −2.12969378601761525102702733354, −0.45137993742406477461521118847,
2.22636567632915624930964306813, 3.72159651644478155433840539117, 5.13971427684165479323116630630, 5.91996185243801279934354378099, 6.50459586337627553074793699440, 7.80801107900961431314191209684, 8.019389049208520714786536527266, 9.122481737456502235815765675289, 10.19112477978328057963060988644, 10.74139258497907352483718426274
