| L(s) = 1 | + (−0.366 + 0.802i)2-s + (0.817 + 0.240i)3-s + (0.799 + 0.923i)4-s + (0.841 − 0.540i)5-s + (−0.492 + 0.568i)6-s + (0.192 − 1.33i)7-s + (−2.72 + 0.800i)8-s + (−1.91 − 1.22i)9-s + (0.125 + 0.873i)10-s + (2.43 + 5.33i)11-s + (0.432 + 0.946i)12-s + (−0.441 − 3.07i)13-s + (1.00 + 0.645i)14-s + (0.817 − 0.240i)15-s + (0.00922 − 0.0641i)16-s + (−0.255 + 0.294i)17-s + ⋯ |
| L(s) = 1 | + (−0.259 + 0.567i)2-s + (0.471 + 0.138i)3-s + (0.399 + 0.461i)4-s + (0.376 − 0.241i)5-s + (−0.200 + 0.231i)6-s + (0.0727 − 0.506i)7-s + (−0.964 + 0.283i)8-s + (−0.637 − 0.409i)9-s + (0.0397 + 0.276i)10-s + (0.734 + 1.60i)11-s + (0.124 + 0.273i)12-s + (−0.122 − 0.851i)13-s + (0.268 + 0.172i)14-s + (0.211 − 0.0619i)15-s + (0.00230 − 0.0160i)16-s + (−0.0618 + 0.0713i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01375 + 0.543279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01375 + 0.543279i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (2.42 + 4.13i)T \) |
| good | 2 | \( 1 + (0.366 - 0.802i)T + (-1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.817 - 0.240i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.192 + 1.33i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.43 - 5.33i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.441 + 3.07i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.255 - 0.294i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.98 + 3.45i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (0.942 - 1.08i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-9.51 + 2.79i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (2.91 + 1.87i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (5.78 - 3.72i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.0590 - 0.0173i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 + (2.00 - 13.9i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.0218 - 0.152i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (4.20 - 1.23i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (5.59 - 12.2i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.69 + 5.90i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (3.21 + 3.70i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.13 - 7.90i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-7.23 - 4.65i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (13.9 + 4.10i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-4.05 + 2.60i)T + (40.2 - 88.2i)T^{2} \) |
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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
−13.91959170933253166571113208624, −12.62686082350114718403409909664, −11.87514868387297463705215653345, −10.38868590197776421593764599771, −9.246460928509977913298939615609, −8.319941874317508488856526411337, −7.19207945693505452468471209186, −6.16106491016882887400177524662, −4.33382317883277215360064582105, −2.60472081901400951804327955766,
1.95185132631726096968058410299, 3.31141416719922340302792511388, 5.66105742233760701681674067113, 6.52700809428152162009292210839, 8.362661448229859350896014563551, 9.140115805554623792370196356922, 10.33049870942521229315849777895, 11.39030788666954292024203026944, 11.98048802439010201226727267687, 13.72640096920863430000842605451
