| L(s) = 1 | + (−0.0402 − 0.999i)2-s + (−2.16 + 0.923i)3-s + (−0.996 + 0.0804i)4-s + (1.34 + 0.218i)5-s + (1.01 + 2.12i)6-s + (2.82 + 2.12i)7-s + (0.120 + 0.992i)8-s + (1.77 − 1.84i)9-s + (0.163 − 1.34i)10-s + (1.85 + 2.27i)11-s + (2.08 − 1.09i)12-s + (1.71 − 3.61i)13-s + (2.00 − 2.91i)14-s + (−3.11 + 0.766i)15-s + (0.987 − 0.160i)16-s + (3.18 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.0284 − 0.706i)2-s + (−1.25 + 0.533i)3-s + (−0.498 + 0.0402i)4-s + (0.600 + 0.0975i)5-s + (0.412 + 0.869i)6-s + (1.06 + 0.802i)7-s + (0.0426 + 0.350i)8-s + (0.590 − 0.614i)9-s + (0.0518 − 0.426i)10-s + (0.560 + 0.685i)11-s + (0.602 − 0.316i)12-s + (0.475 − 1.00i)13-s + (0.536 − 0.777i)14-s + (−0.803 + 0.198i)15-s + (0.246 − 0.0401i)16-s + (0.771 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.876011 + 0.0878154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.876011 + 0.0878154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0402 + 0.999i)T \) |
| 79 | \( 1 + (3.48 + 8.17i)T \) |
| good | 3 | \( 1 + (2.16 - 0.923i)T + (2.07 - 2.16i)T^{2} \) |
| 5 | \( 1 + (-1.34 - 0.218i)T + (4.74 + 1.58i)T^{2} \) |
| 7 | \( 1 + (-2.82 - 2.12i)T + (1.94 + 6.72i)T^{2} \) |
| 11 | \( 1 + (-1.85 - 2.27i)T + (-2.20 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.71 + 3.61i)T + (-8.22 - 10.0i)T^{2} \) |
| 17 | \( 1 + (-3.18 - 4.60i)T + (-6.02 + 15.8i)T^{2} \) |
| 19 | \( 1 + (1.32 - 0.442i)T + (15.1 - 11.4i)T^{2} \) |
| 23 | \( 1 + (2.66 - 4.62i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.675 + 2.33i)T + (-24.5 - 15.4i)T^{2} \) |
| 31 | \( 1 + (4.23 - 2.67i)T + (13.2 - 28.0i)T^{2} \) |
| 37 | \( 1 + (-0.640 + 3.13i)T + (-34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (3.00 + 7.92i)T + (-30.6 + 27.1i)T^{2} \) |
| 43 | \( 1 + (-7.57 + 9.27i)T + (-8.60 - 42.1i)T^{2} \) |
| 47 | \( 1 + (-1.48 - 7.27i)T + (-43.2 + 18.4i)T^{2} \) |
| 53 | \( 1 + (-6.98 - 2.97i)T + (36.7 + 38.2i)T^{2} \) |
| 59 | \( 1 + (9.71 + 0.784i)T + (58.2 + 9.46i)T^{2} \) |
| 61 | \( 1 + (-2.11 - 1.87i)T + (7.35 + 60.5i)T^{2} \) |
| 67 | \( 1 + (-7.90 + 4.14i)T + (38.0 - 55.1i)T^{2} \) |
| 71 | \( 1 + (1.07 + 8.89i)T + (-68.9 + 16.9i)T^{2} \) |
| 73 | \( 1 + (5.95 + 12.5i)T + (-46.1 + 56.5i)T^{2} \) |
| 83 | \( 1 + (9.55 - 0.771i)T + (81.9 - 13.3i)T^{2} \) |
| 89 | \( 1 + (1.70 - 14.0i)T + (-86.4 - 21.2i)T^{2} \) |
| 97 | \( 1 + (-9.77 - 8.66i)T + (11.6 + 96.2i)T^{2} \) |
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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
−12.39916217421113171173976015609, −11.96983680293845000612724255373, −10.83672003998196599125520837000, −10.33391299469378499374698535537, −9.173621509716499308286201805854, −7.901015938834097096190375105695, −5.88752106386726056609536022830, −5.42116315557044709448758839572, −4.02403144776240411120242267460, −1.82299216245693364988866064956,
1.20715757239748980725155701912, 4.34137673266231493355116388371, 5.50330563084286007315387044912, 6.40833768296407608863819787994, 7.32650381834369429001640877790, 8.575294486477032870288730513044, 9.876749680293233650683486844023, 11.18649525732263744658109555761, 11.66560802867659595103835668908, 12.99119638198902416343335726833
