| L(s) = 1 | + (0.190 + 2.65i)2-s + (2.92 + 0.637i)3-s + (−3.07 + 0.441i)4-s + (4.85 + 1.19i)5-s + (−1.13 + 7.90i)6-s + (3.40 − 6.23i)7-s + (0.508 + 2.33i)8-s + (−0.0154 − 0.00705i)9-s + (−2.24 + 13.1i)10-s + (−11.3 − 13.1i)11-s + (−9.27 − 0.663i)12-s + (6.58 + 12.0i)13-s + (17.2 + 7.85i)14-s + (13.4 + 6.58i)15-s + (−18.0 + 5.29i)16-s + (−14.9 − 20.0i)17-s + ⋯ |
| L(s) = 1 | + (0.0950 + 1.32i)2-s + (0.976 + 0.212i)3-s + (−0.767 + 0.110i)4-s + (0.971 + 0.238i)5-s + (−0.189 + 1.31i)6-s + (0.486 − 0.890i)7-s + (0.0636 + 0.292i)8-s + (−0.00171 − 0.000783i)9-s + (−0.224 + 1.31i)10-s + (−1.03 − 1.19i)11-s + (−0.772 − 0.0552i)12-s + (0.506 + 0.927i)13-s + (1.22 + 0.561i)14-s + (0.897 + 0.438i)15-s + (−1.12 + 0.330i)16-s + (−0.881 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.40473 + 1.57301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40473 + 1.57301i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.85 - 1.19i)T \) |
| 23 | \( 1 + (22.5 - 4.67i)T \) |
| good | 2 | \( 1 + (-0.190 - 2.65i)T + (-3.95 + 0.569i)T^{2} \) |
| 3 | \( 1 + (-2.92 - 0.637i)T + (8.18 + 3.73i)T^{2} \) |
| 7 | \( 1 + (-3.40 + 6.23i)T + (-26.4 - 41.2i)T^{2} \) |
| 11 | \( 1 + (11.3 + 13.1i)T + (-17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (-6.58 - 12.0i)T + (-91.3 + 142. i)T^{2} \) |
| 17 | \( 1 + (14.9 + 20.0i)T + (-81.4 + 277. i)T^{2} \) |
| 19 | \( 1 + (25.4 - 3.66i)T + (346. - 101. i)T^{2} \) |
| 29 | \( 1 + (-32.0 - 4.60i)T + (806. + 236. i)T^{2} \) |
| 31 | \( 1 + (-42.8 - 27.5i)T + (399. + 874. i)T^{2} \) |
| 37 | \( 1 + (-6.50 - 17.4i)T + (-1.03e3 + 896. i)T^{2} \) |
| 41 | \( 1 + (0.871 + 1.90i)T + (-1.10e3 + 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-21.2 - 4.62i)T + (1.68e3 + 768. i)T^{2} \) |
| 47 | \( 1 + (-36.1 + 36.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-7.30 - 3.99i)T + (1.51e3 + 2.36e3i)T^{2} \) |
| 59 | \( 1 + (-5.48 + 18.6i)T + (-2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (-17.0 - 10.9i)T + (1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (-1.88 - 26.4i)T + (-4.44e3 + 638. i)T^{2} \) |
| 71 | \( 1 + (84.0 - 97.0i)T + (-717. - 4.98e3i)T^{2} \) |
| 73 | \( 1 + (7.57 - 10.1i)T + (-1.50e3 - 5.11e3i)T^{2} \) |
| 79 | \( 1 + (-4.19 + 14.2i)T + (-5.25e3 - 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-10.8 + 4.02i)T + (5.20e3 - 4.51e3i)T^{2} \) |
| 89 | \( 1 + (-25.4 - 39.5i)T + (-3.29e3 + 7.20e3i)T^{2} \) |
| 97 | \( 1 + (96.3 + 35.9i)T + (7.11e3 + 6.16e3i)T^{2} \) |
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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
−13.75813216610696798803906037819, −13.61640333181519517914542300866, −11.30046104034600967787732666165, −10.28013317985434871782276066818, −8.792829727998810892419074680079, −8.274565955692483871408183588072, −6.92116073345891750430855803717, −5.94668304042991401742785154504, −4.49318004580291177185587871817, −2.54299439050681300267095575086,
2.09455077610475568406737350970, 2.50989453414499834183794075300, 4.48908810227944282916462871184, 6.09081115810241540672878100651, 8.080058381824359897479653851963, 8.879369304641292904064392853637, 10.10464846026742406199656738390, 10.76889508603759597134732976456, 12.28768750279412162618866363851, 12.95990048646369946358634318733
