| L(s) = 1 | + (0.347 + 1.96i)2-s + (−3.75 + 1.36i)4-s + (14.9 + 12.5i)5-s + (11.7 + 4.28i)7-s + (−4 − 6.92i)8-s + (−19.4 + 33.7i)10-s + (9.63 − 8.08i)11-s + (6.96 − 39.5i)13-s + (−4.35 + 24.6i)14-s + (12.2 − 10.2i)16-s + (−24.5 + 42.5i)17-s + (67.8 + 117. i)19-s + (−73.1 − 26.6i)20-s + (19.2 + 16.1i)22-s + (−162. + 59.3i)23-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.469 + 0.171i)4-s + (1.33 + 1.11i)5-s + (0.635 + 0.231i)7-s + (−0.176 − 0.306i)8-s + (−0.615 + 1.06i)10-s + (0.263 − 0.221i)11-s + (0.148 − 0.842i)13-s + (−0.0830 + 0.471i)14-s + (0.191 − 0.160i)16-s + (−0.350 + 0.606i)17-s + (0.819 + 1.41i)19-s + (−0.817 − 0.297i)20-s + (0.186 + 0.156i)22-s + (−1.47 + 0.537i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.35318 + 1.74136i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35318 + 1.74136i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 - 1.96i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-14.9 - 12.5i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-11.7 - 4.28i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (-9.63 + 8.08i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-6.96 + 39.5i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (24.5 - 42.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-67.8 - 117. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (162. - 59.3i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (7.51 + 42.6i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (18.2 - 6.63i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-124. + 215. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-68.5 + 389. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (164. - 138. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (267. + 97.3i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-320. - 269. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-78.1 - 28.4i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-104. + 595. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (25.4 - 44.0i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-444. - 770. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (29.3 + 166. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (257. + 1.45e3i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (401. + 694. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.14e3 + 957. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.99800430101887423770992239693, −11.67081310161808665193170593878, −10.43237465298953078171499975693, −9.775078394819717622652733100498, −8.411927095510624776174794214978, −7.35588536707645727998376656832, −6.02507216728052738827803073889, −5.59162253285337822621055809320, −3.62411239174923304232573141836, −1.95219339050500291382763717569,
1.12197877113604127251343018984, 2.28983611491994075841017987835, 4.43239238075454314413067265074, 5.19585470630571639703443765479, 6.57565903084908547079155688280, 8.326731575249144884987362061859, 9.319760057860909601195855958586, 9.889176860304609931771775117769, 11.26924955675295214190632426600, 12.06103094752996513420797046110
