| L(s) = 1 | + (4.54 + 3.36i)2-s + 6.93·3-s + (9.39 + 30.5i)4-s + (31.5 + 23.2i)6-s − 47.1i·7-s + (−60.0 + 170. i)8-s − 194.·9-s + 253. i·11-s + (65.1 + 212. i)12-s − 1.03e3·13-s + (158. − 214. i)14-s + (−847. + 574. i)16-s − 756. i·17-s + (−887. − 655. i)18-s + 344. i·19-s + ⋯ |
| L(s) = 1 | + (0.804 + 0.594i)2-s + 0.444·3-s + (0.293 + 0.955i)4-s + (0.357 + 0.264i)6-s − 0.363i·7-s + (−0.331 + 0.943i)8-s − 0.802·9-s + 0.632i·11-s + (0.130 + 0.425i)12-s − 1.69·13-s + (0.216 − 0.292i)14-s + (−0.827 + 0.561i)16-s − 0.634i·17-s + (−0.645 − 0.476i)18-s + 0.218i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.541259264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541259264\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.54 - 3.36i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 6.93T + 243T^{2} \) |
| 7 | \( 1 + 47.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 253. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 756. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 344. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.97e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 372. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 134.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.59e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.77e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.40e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.89e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.01e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.31e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.12e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.25e4iT - 8.58e9T^{2} \) |
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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
−12.11659605496231974164844824485, −11.53976388290373429673794428226, −9.990525633243030625850004088021, −8.973547954430574664458859006357, −7.65739717130770547604526741317, −7.18155986507980806559716389008, −5.66641444476843494537734485730, −4.73064276003877267977507104865, −3.39055734256487321664211215835, −2.26427191144235684764500242314,
0.29848162933361146053598221286, 2.24318295900607132139417139590, 3.02475528467102847077495584633, 4.48353076956724964476025082208, 5.57656705669343796256759217330, 6.65927885111299633385384404409, 8.155359694173539007090891674680, 9.206994021272754282976006083421, 10.24457610045089186091700386362, 11.22879789924263038951598933382
