| L(s) = 1 | + (−4.96 + 0.283i)2-s + (−0.441 − 1.35i)3-s + (16.5 − 1.90i)4-s + (−1.29 + 2.45i)5-s + (2.57 + 6.60i)6-s + (−18.6 + 7.90i)7-s + (−42.5 + 7.37i)8-s + (20.1 − 14.6i)9-s + (5.71 − 12.5i)10-s + (15.4 − 33.0i)11-s + (−9.89 − 21.6i)12-s + (0.348 + 0.196i)13-s + (90.5 − 44.5i)14-s + (3.89 + 0.674i)15-s + (79.1 − 18.4i)16-s + (−52.3 − 18.6i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 + 0.100i)2-s + (−0.0848 − 0.261i)3-s + (2.07 − 0.237i)4-s + (−0.115 + 0.219i)5-s + (0.175 + 0.449i)6-s + (−1.00 + 0.426i)7-s + (−1.88 + 0.325i)8-s + (0.747 − 0.543i)9-s + (0.180 − 0.396i)10-s + (0.424 − 0.905i)11-s + (−0.238 − 0.521i)12-s + (0.00742 + 0.00419i)13-s + (1.72 − 0.849i)14-s + (0.0670 + 0.0116i)15-s + (1.23 − 0.287i)16-s + (−0.746 − 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.214046 + 0.277820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.214046 + 0.277820i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-15.4 + 33.0i)T \) |
| good | 2 | \( 1 + (4.96 - 0.283i)T + (7.94 - 0.911i)T^{2} \) |
| 3 | \( 1 + (0.441 + 1.35i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (1.29 - 2.45i)T + (-70.5 - 103. i)T^{2} \) |
| 7 | \( 1 + (18.6 - 7.90i)T + (239. - 245. i)T^{2} \) |
| 13 | \( 1 + (-0.348 - 0.196i)T + (1.13e3 + 1.88e3i)T^{2} \) |
| 17 | \( 1 + (52.3 + 18.6i)T + (3.80e3 + 3.10e3i)T^{2} \) |
| 19 | \( 1 + (-3.82 - 133. i)T + (-6.84e3 + 391. i)T^{2} \) |
| 23 | \( 1 + (56.4 - 65.1i)T + (-1.73e3 - 1.20e4i)T^{2} \) |
| 29 | \( 1 + (52.6 - 76.9i)T + (-8.84e3 - 2.27e4i)T^{2} \) |
| 31 | \( 1 + (-39.9 - 197. i)T + (-2.74e4 + 1.15e4i)T^{2} \) |
| 37 | \( 1 + (38.3 + 35.2i)T + (4.33e3 + 5.04e4i)T^{2} \) |
| 41 | \( 1 + (30.7 - 117. i)T + (-6.00e4 - 3.38e4i)T^{2} \) |
| 43 | \( 1 + (70.2 - 488. i)T + (-7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 + (-132. - 108. i)T + (2.06e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (465. + 108. i)T + (1.33e5 + 6.56e4i)T^{2} \) |
| 59 | \( 1 + (94.5 + 359. i)T + (-1.78e5 + 1.00e5i)T^{2} \) |
| 61 | \( 1 + (-208. - 11.9i)T + (2.25e5 + 2.58e4i)T^{2} \) |
| 67 | \( 1 + (-595. - 382. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-230. - 299. i)T + (-9.09e4 + 3.46e5i)T^{2} \) |
| 73 | \( 1 + (368. + 611. i)T + (-1.81e5 + 3.44e5i)T^{2} \) |
| 79 | \( 1 + (781. + 803. i)T + (-1.40e4 + 4.92e5i)T^{2} \) |
| 83 | \( 1 + (-71.6 - 834. i)T + (-5.63e5 + 9.75e4i)T^{2} \) |
| 89 | \( 1 + (-352. + 103. i)T + (5.93e5 - 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-188. - 357. i)T + (-5.15e5 + 7.53e5i)T^{2} \) |
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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
−13.00060345257259140163142461944, −11.95295637248943873126371182228, −10.92915445127872772850279928999, −9.818311375371012591679280542269, −9.178441504933240419953212436888, −8.050757808632776522161447854045, −6.85972529011335943901266315298, −6.13942555207329589619822600800, −3.32724503107493848054616837428, −1.38589077666690034286129069762,
0.34468913450180936067690729434, 2.22019460313226294788495980006, 4.35481225858057165294576316230, 6.63193043396455433929651407908, 7.29871780854464392449306368716, 8.637901672284499368225729356801, 9.611135572378529746890147307495, 10.23371300273701964076303487102, 11.18172450177978842869069525835, 12.44244232612131985059132710590
