| L(s) = 1 | + (−2.82 − 4.36i)3-s + (9.22 − 15.9i)5-s + (−6.70 − 17.2i)7-s + (−11.0 + 24.6i)9-s + (11.8 − 6.86i)11-s − 9.57i·13-s + (−95.7 + 4.87i)15-s + (48.2 + 83.5i)17-s + (−23.2 − 13.4i)19-s + (−56.3 + 77.9i)21-s + (−153. − 88.8i)23-s + (−107. − 186. i)25-s + (138. − 21.2i)27-s − 131. i·29-s + (−273. + 157. i)31-s + ⋯ |
| L(s) = 1 | + (−0.543 − 0.839i)3-s + (0.825 − 1.42i)5-s + (−0.361 − 0.932i)7-s + (−0.409 + 0.912i)9-s + (0.326 − 0.188i)11-s − 0.204i·13-s + (−1.64 + 0.0838i)15-s + (0.688 + 1.19i)17-s + (−0.280 − 0.162i)19-s + (−0.585 + 0.810i)21-s + (−1.39 − 0.805i)23-s + (−0.862 − 1.49i)25-s + (0.988 − 0.151i)27-s − 0.840i·29-s + (−1.58 + 0.914i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.181622 - 1.22494i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.181622 - 1.22494i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 + 4.36i)T \) |
| 7 | \( 1 + (6.70 + 17.2i)T \) |
| good | 5 | \( 1 + (-9.22 + 15.9i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-11.8 + 6.86i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 9.57iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-48.2 - 83.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (23.2 + 13.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (153. + 88.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 131. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (273. - 157. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (54.7 - 94.7i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 390.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-149. + 258. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-6.68 + 3.85i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (133. + 231. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-195. - 113. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-296. - 512. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 864. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-474. + 274. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-91.1 + 157. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 54.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (403. - 698. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.42e3iT - 9.12e5T^{2} \) |
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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
−12.35594374904880753996042033842, −10.85161473849969243243776435518, −9.934028290535760738261852870127, −8.690450521244559075226874368355, −7.73242599772949688426435204926, −6.33428354539858725284879271093, −5.55538696780491448549651902946, −4.17143127807582390829893221995, −1.79507533110085402731431815518, −0.60951939030728961891305917400,
2.42946783089870700479478687633, 3.68810684581873231822073686002, 5.51938641995996651860755942271, 6.13520264830620309506429105704, 7.32858101780660589216931881622, 9.295298608579034024538262193664, 9.687322762665125214775747174264, 10.77611161578508813356392701320, 11.56653296066148912302993200904, 12.57573631466669546199255513892
