| L(s) = 1 | + (0.371 + 0.259i)2-s + (−3.02 − 0.264i)3-s + (−0.613 − 1.68i)4-s + (−2.23 + 0.122i)5-s + (−1.05 − 0.884i)6-s + (−1.89 + 0.508i)7-s + (0.444 − 1.66i)8-s + (6.13 + 1.08i)9-s + (−0.860 − 0.534i)10-s + (0.238 + 0.412i)11-s + (1.41 + 5.26i)12-s + (−0.364 − 4.16i)13-s + (−0.836 − 0.304i)14-s + (6.79 + 0.219i)15-s + (−2.15 + 1.80i)16-s + (−0.463 + 0.661i)17-s + ⋯ |
| L(s) = 1 | + (0.262 + 0.183i)2-s + (−1.74 − 0.152i)3-s + (−0.306 − 0.843i)4-s + (−0.998 + 0.0549i)5-s + (−0.430 − 0.361i)6-s + (−0.717 + 0.192i)7-s + (0.157 − 0.587i)8-s + (2.04 + 0.360i)9-s + (−0.272 − 0.169i)10-s + (0.0717 + 0.124i)11-s + (0.407 + 1.52i)12-s + (−0.101 − 1.15i)13-s + (−0.223 − 0.0813i)14-s + (1.75 + 0.0566i)15-s + (−0.538 + 0.451i)16-s + (−0.112 + 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0691434 - 0.234981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0691434 - 0.234981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.23 - 0.122i)T \) |
| 19 | \( 1 + (2.59 + 3.49i)T \) |
| good | 2 | \( 1 + (-0.371 - 0.259i)T + (0.684 + 1.87i)T^{2} \) |
| 3 | \( 1 + (3.02 + 0.264i)T + (2.95 + 0.520i)T^{2} \) |
| 7 | \( 1 + (1.89 - 0.508i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.238 - 0.412i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.364 + 4.16i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (0.463 - 0.661i)T + (-5.81 - 15.9i)T^{2} \) |
| 23 | \( 1 + (1.97 - 4.23i)T + (-14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.693 + 3.93i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (8.78 + 5.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.48 + 4.48i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.84 - 5.77i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.992 + 0.462i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-7.61 + 5.33i)T + (16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (6.51 + 3.03i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.304 - 1.72i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.30 - 1.20i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.74 + 2.48i)T + (-22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (0.907 - 2.49i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.0344 + 0.394i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (-3.51 + 2.95i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.03 - 7.60i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.225 + 0.189i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.38 - 2.37i)T + (33.1 + 91.1i)T^{2} \) |
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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
−13.15536696220456428020163642239, −12.55730478719834662820982550027, −11.33948264579364915584616287542, −10.65665499253979652636778387275, −9.510529278335987307218814037124, −7.49725032890805686555993497660, −6.31425274287699602574035252681, −5.46229264854739773580428427893, −4.20711316453361540309826138924, −0.32833454403412219959521774606,
3.83185910161799838516766292125, 4.74539039350521280764322317876, 6.38745259454021127745560834948, 7.41731859145687288269648131440, 9.014861811809624918022617145923, 10.60093160722213136090049874555, 11.43300028168310887617554663469, 12.34318106698798955820295042541, 12.71550579001385182996831055243, 14.33286144089729724979620092038
