| L(s) = 1 | + (−0.287 + 0.166i)2-s + (−0.729 + 1.26i)3-s + (−0.944 + 1.63i)4-s + (−1.25 + 0.722i)5-s − 0.485i·6-s + (−2.26 − 1.36i)7-s − 1.29i·8-s + (0.434 + 0.752i)9-s + (0.240 − 0.416i)10-s + (5.15 + 2.97i)11-s + (−1.37 − 2.38i)12-s + (1.88 + 3.07i)13-s + (0.879 + 0.0178i)14-s − 2.11i·15-s + (−1.67 − 2.90i)16-s + (2.16 − 3.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.203 + 0.117i)2-s + (−0.421 + 0.729i)3-s + (−0.472 + 0.818i)4-s + (−0.559 + 0.323i)5-s − 0.198i·6-s + (−0.855 − 0.517i)7-s − 0.457i·8-s + (0.144 + 0.250i)9-s + (0.0759 − 0.131i)10-s + (1.55 + 0.897i)11-s + (−0.398 − 0.689i)12-s + (0.524 + 0.851i)13-s + (0.234 + 0.00477i)14-s − 0.544i·15-s + (−0.418 − 0.725i)16-s + (0.524 − 0.909i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.327949 + 0.533162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.327949 + 0.533162i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.26 + 1.36i)T \) |
| 13 | \( 1 + (-1.88 - 3.07i)T \) |
| good | 2 | \( 1 + (0.287 - 0.166i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.729 - 1.26i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.25 - 0.722i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.15 - 2.97i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 3.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.69 - 0.978i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.270 + 0.467i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 + (5.28 + 3.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.95 + 4.01i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.55iT - 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + (-5.42 + 3.13i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.38 + 2.40i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.737 - 0.425i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.38 + 5.87i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.854 + 0.493i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.76iT - 71T^{2} \) |
| 73 | \( 1 + (-7.91 - 4.56i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0655 - 0.113i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.66iT - 83T^{2} \) |
| 89 | \( 1 + (-8.41 + 4.85i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.58iT - 97T^{2} \) |
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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
−14.46272136008758186971020841768, −13.36073715427898994230476379753, −12.16419820404556397572983747519, −11.31069651555224579824571088784, −9.863314128251826168816946781503, −9.209984742982313074951022900272, −7.57334326538579638256809549523, −6.62571673564314392268318103051, −4.43059854632253306029957502812, −3.70677513696973079012478685328,
0.964227888640468369760857581907, 3.82651212048661204288555503922, 5.84093105180605494087032141823, 6.48628070345805857111518212720, 8.364125922707267870142610900369, 9.244078615350286252565623463698, 10.52339582705287532156831761802, 11.79598137203079043214311127976, 12.56193058491671707736548168657, 13.61246559949443063989164083446
