| L(s) = 1 | + (0.830 + 1.81i)2-s + (6.99 − 2.05i)3-s + (−2.61 + 3.02i)4-s + (2.74 + 1.76i)5-s + (9.55 + 11.0i)6-s + (−1.74 − 12.1i)7-s + (−7.67 − 2.25i)8-s + (22.0 − 14.1i)9-s + (−0.928 + 6.45i)10-s + (−20.1 + 44.1i)11-s + (−12.1 + 26.5i)12-s + (0.229 − 1.59i)13-s + (20.6 − 13.2i)14-s + (22.8 + 6.70i)15-s + (−2.27 − 15.8i)16-s + (−58.0 − 67.0i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (1.34 − 0.395i)3-s + (−0.327 + 0.377i)4-s + (0.245 + 0.157i)5-s + (0.649 + 0.749i)6-s + (−0.0943 − 0.656i)7-s + (−0.339 − 0.0996i)8-s + (0.815 − 0.524i)9-s + (−0.0293 + 0.204i)10-s + (−0.552 + 1.21i)11-s + (−0.291 + 0.638i)12-s + (0.00490 − 0.0341i)13-s + (0.394 − 0.253i)14-s + (0.392 + 0.115i)15-s + (−0.0355 − 0.247i)16-s + (−0.828 − 0.956i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.00415 + 0.567624i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00415 + 0.567624i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.830 - 1.81i)T \) |
| 23 | \( 1 + (46.4 + 100. i)T \) |
| good | 3 | \( 1 + (-6.99 + 2.05i)T + (22.7 - 14.5i)T^{2} \) |
| 5 | \( 1 + (-2.74 - 1.76i)T + (51.9 + 113. i)T^{2} \) |
| 7 | \( 1 + (1.74 + 12.1i)T + (-329. + 96.6i)T^{2} \) |
| 11 | \( 1 + (20.1 - 44.1i)T + (-871. - 1.00e3i)T^{2} \) |
| 13 | \( 1 + (-0.229 + 1.59i)T + (-2.10e3 - 618. i)T^{2} \) |
| 17 | \( 1 + (58.0 + 67.0i)T + (-699. + 4.86e3i)T^{2} \) |
| 19 | \( 1 + (-7.24 + 8.35i)T + (-976. - 6.78e3i)T^{2} \) |
| 29 | \( 1 + (-182. - 210. i)T + (-3.47e3 + 2.41e4i)T^{2} \) |
| 31 | \( 1 + (-19.1 - 5.61i)T + (2.50e4 + 1.61e4i)T^{2} \) |
| 37 | \( 1 + (-10.5 + 6.74i)T + (2.10e4 - 4.60e4i)T^{2} \) |
| 41 | \( 1 + (-254. - 163. i)T + (2.86e4 + 6.26e4i)T^{2} \) |
| 43 | \( 1 + (442. - 129. i)T + (6.68e4 - 4.29e4i)T^{2} \) |
| 47 | \( 1 - 508.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-90.4 - 628. i)T + (-1.42e5 + 4.19e4i)T^{2} \) |
| 59 | \( 1 + (-21.7 + 151. i)T + (-1.97e5 - 5.78e4i)T^{2} \) |
| 61 | \( 1 + (-326. - 95.7i)T + (1.90e5 + 1.22e5i)T^{2} \) |
| 67 | \( 1 + (320. + 702. i)T + (-1.96e5 + 2.27e5i)T^{2} \) |
| 71 | \( 1 + (144. + 316. i)T + (-2.34e5 + 2.70e5i)T^{2} \) |
| 73 | \( 1 + (333. - 384. i)T + (-5.53e4 - 3.85e5i)T^{2} \) |
| 79 | \( 1 + (-29.2 + 203. i)T + (-4.73e5 - 1.38e5i)T^{2} \) |
| 83 | \( 1 + (-188. + 120. i)T + (2.37e5 - 5.20e5i)T^{2} \) |
| 89 | \( 1 + (-1.10e3 + 324. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (620. + 398. i)T + (3.79e5 + 8.30e5i)T^{2} \) |
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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
−15.14063038435974291225276991462, −14.15611037655871894297614357001, −13.50220054674143495429318325114, −12.40202376368362884306155649714, −10.27945298393340190988635837588, −8.981142146069239893107739511329, −7.73369213254003088471976442238, −6.79684533602069632239040226902, −4.52870805971276538192290341504, −2.61736652333102304931465292341,
2.40480013994097880154032317988, 3.78826621336371710697004274137, 5.75484422240148460133857444588, 8.225373820720839670121764773035, 9.054745248248703411554949393168, 10.24050036042824894709231352166, 11.66535836566901125665864905776, 13.23532182004806310141573052823, 13.80875565054489982323593559925, 15.07334048699717918631775873178
