| L(s) = 1 | + (−1.18 − 0.528i)2-s + (0.262 − 2.49i)3-s + (−0.209 − 0.232i)4-s − 0.0704·5-s + (−1.63 + 2.82i)6-s + (−3.21 − 3.57i)7-s + (0.928 + 2.85i)8-s + (−3.24 − 0.689i)9-s + (0.0835 + 0.0372i)10-s + (−1.13 + 0.241i)11-s + (−0.635 + 0.461i)12-s + (−1.16 + 3.41i)13-s + (1.92 + 5.93i)14-s + (−0.0184 + 0.175i)15-s + (0.342 − 3.25i)16-s + (3.33 + 0.709i)17-s + ⋯ |
| L(s) = 1 | + (−0.839 − 0.373i)2-s + (0.151 − 1.44i)3-s + (−0.104 − 0.116i)4-s − 0.0314·5-s + (−0.666 + 1.15i)6-s + (−1.21 − 1.35i)7-s + (0.328 + 1.01i)8-s + (−1.08 − 0.229i)9-s + (0.0264 + 0.0117i)10-s + (−0.342 + 0.0728i)11-s + (−0.183 + 0.133i)12-s + (−0.324 + 0.945i)13-s + (0.515 + 1.58i)14-s + (−0.00477 + 0.0454i)15-s + (0.0856 − 0.814i)16-s + (0.809 + 0.172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.204584 + 0.315543i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.204584 + 0.315543i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (1.16 - 3.41i)T \) |
| 31 | \( 1 + (-1.97 - 5.20i)T \) |
| good | 2 | \( 1 + (1.18 + 0.528i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.262 + 2.49i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + 0.0704T + 5T^{2} \) |
| 7 | \( 1 + (3.21 + 3.57i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (1.13 - 0.241i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (-3.33 - 0.709i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.450 + 4.28i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-3.06 + 3.40i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-1.26 - 0.564i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 3.27i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (10.9 + 4.87i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.765 - 7.28i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (1.44 + 1.04i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.40 + 4.31i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (10.8 - 4.85i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-6.27 + 10.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.94 + 3.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.2 + 2.17i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (3.04 - 9.38i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (4.03 + 12.4i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-9.09 + 6.60i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.639 + 0.135i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (2.18 + 2.43i)T + (-10.1 + 96.4i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.42480175555773812160273498529, −9.819156170267409068700976307582, −8.812807518617464114673119783948, −7.79186736678874821726748513986, −7.03817043320919552627889442184, −6.34165614188798291983457726102, −4.68394215533634937421819051926, −3.01022262096875951341829355573, −1.56735289676674014023525388476, −0.32319572215034629491096926394,
2.98264650458194572290708524443, 3.76380148146057094966387126798, 5.24521329962420772972700529852, 6.06345650841494732436739126822, 7.58537695337737244053402293331, 8.482430895625962504205618745246, 9.311963404881497825527591215329, 9.879447461515423186329997569937, 10.32942363134925562823579148236, 11.82858724884312809407489596667
