| L(s) = 1 | + (−0.580 − 3.29i)3-s + (−11.9 + 9.99i)5-s + (−13.7 + 23.8i)7-s + (14.8 − 5.41i)9-s + (17.2 + 29.9i)11-s + (−3.57 + 20.2i)13-s + (39.7 + 33.3i)15-s + (−102. − 37.2i)17-s + (−76.2 + 32.3i)19-s + (86.4 + 31.4i)21-s + (−12.9 − 10.8i)23-s + (20.2 − 114. i)25-s + (−71.5 − 123. i)27-s + (232. − 84.6i)29-s + (−136. + 235. i)31-s + ⋯ |
| L(s) = 1 | + (−0.111 − 0.633i)3-s + (−1.06 + 0.893i)5-s + (−0.743 + 1.28i)7-s + (0.551 − 0.200i)9-s + (0.473 + 0.820i)11-s + (−0.0763 + 0.432i)13-s + (0.684 + 0.574i)15-s + (−1.45 − 0.531i)17-s + (−0.920 + 0.391i)19-s + (0.898 + 0.326i)21-s + (−0.117 − 0.0987i)23-s + (0.161 − 0.918i)25-s + (−0.510 − 0.883i)27-s + (1.48 − 0.541i)29-s + (−0.788 + 1.36i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.421141 + 0.610374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421141 + 0.610374i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (76.2 - 32.3i)T \) |
| good | 3 | \( 1 + (0.580 + 3.29i)T + (-25.3 + 9.23i)T^{2} \) |
| 5 | \( 1 + (11.9 - 9.99i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (13.7 - 23.8i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-17.2 - 29.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (3.57 - 20.2i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (102. + 37.2i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 23 | \( 1 + (12.9 + 10.8i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-232. + 84.6i)T + (1.86e4 - 1.56e4i)T^{2} \) |
| 31 | \( 1 + (136. - 235. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 340.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-3.48 - 19.7i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (61.2 - 51.4i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-14.3 + 5.23i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + (235. + 198. i)T + (2.58e4 + 1.46e5i)T^{2} \) |
| 59 | \( 1 + (-605. - 220. i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-366. - 307. i)T + (3.94e4 + 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-90.9 + 33.1i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (312. - 261. i)T + (6.21e4 - 3.52e5i)T^{2} \) |
| 73 | \( 1 + (-151. - 861. i)T + (-3.65e5 + 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-120. - 682. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (225. - 390. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-131. + 744. i)T + (-6.62e5 - 2.41e5i)T^{2} \) |
| 97 | \( 1 + (314. + 114. i)T + (6.99e5 + 5.86e5i)T^{2} \) |
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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
−14.55867165019320270484467296690, −12.94617519327376112561249519062, −12.18038603384757706169460444565, −11.36580410390222610806486293457, −9.842452204313377926160153529229, −8.561883491817052628319484076098, −7.02825054197731875924717644396, −6.47015133560863919382015575475, −4.22908000635638502120227942219, −2.44949062175615502322889796649,
0.47480089829084692003529168505, 3.85112403285144590173660406094, 4.51488146588137490508030350138, 6.58266846638391932280825061681, 7.951154224367614487980198892195, 9.133887311171721234061466278967, 10.45118612942002696601560916834, 11.27411315491990291049228044929, 12.78830738807403169682614313773, 13.39850988849194699905543322086
