| 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
−12.44244232612131985059132710590, −11.18172450177978842869069525835, −10.23371300273701964076303487102, −9.611135572378529746890147307495, −8.637901672284499368225729356801, −7.29871780854464392449306368716, −6.63193043396455433929651407908, −4.35481225858057165294576316230, −2.22019460313226294788495980006, −0.34468913450180936067690729434,
1.38589077666690034286129069762, 3.32724503107493848054616837428, 6.13942555207329589619822600800, 6.85972529011335943901266315298, 8.050757808632776522161447854045, 9.178441504933240419953212436888, 9.818311375371012591679280542269, 10.92915445127872772850279928999, 11.95295637248943873126371182228, 13.00060345257259140163142461944
