| L(s) = 1 | + (1.40 + 0.143i)2-s + (0.0395 + 0.0180i)3-s + (1.95 + 0.403i)4-s + (0.0591 − 0.201i)5-s + (0.0530 + 0.0311i)6-s + (0.929 − 2.68i)7-s + (2.69 + 0.848i)8-s + (−1.96 − 2.26i)9-s + (0.112 − 0.274i)10-s + (3.37 − 0.819i)11-s + (0.0702 + 0.0513i)12-s + (−0.0742 − 0.143i)13-s + (1.69 − 3.64i)14-s + (0.00598 − 0.00690i)15-s + (3.67 + 1.58i)16-s + (−1.05 − 0.420i)17-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.101i)2-s + (0.0228 + 0.0104i)3-s + (0.979 + 0.201i)4-s + (0.0264 − 0.0900i)5-s + (0.0216 + 0.0126i)6-s + (0.351 − 1.01i)7-s + (0.953 + 0.300i)8-s + (−0.654 − 0.755i)9-s + (0.0354 − 0.0869i)10-s + (1.01 − 0.247i)11-s + (0.0202 + 0.0148i)12-s + (−0.0205 − 0.0399i)13-s + (0.452 − 0.974i)14-s + (0.00154 − 0.00178i)15-s + (0.918 + 0.395i)16-s + (−0.254 − 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.74277 - 0.444270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.74277 - 0.444270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.143i)T \) |
| 67 | \( 1 + (5.52 + 6.03i)T \) |
| good | 3 | \( 1 + (-0.0395 - 0.0180i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.0591 + 0.201i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.929 + 2.68i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-3.37 + 0.819i)T + (9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (0.0742 + 0.143i)T + (-7.54 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.05 + 0.420i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (7.33 - 2.53i)T + (14.9 - 11.7i)T^{2} \) |
| 23 | \( 1 + (-4.30 - 6.04i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 0.755i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.20 + 0.622i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (-4.06 - 2.34i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.288 - 0.227i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (11.3 - 1.63i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (8.95 - 0.854i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (3.59 + 0.516i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (4.20 + 6.53i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-9.26 - 2.24i)T + (54.2 + 27.9i)T^{2} \) |
| 71 | \( 1 + (-3.25 + 1.30i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (0.642 - 2.64i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.537 - 11.2i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (6.02 + 6.31i)T + (-3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (3.01 + 6.60i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-1.91 + 3.32i)T + (-48.5 - 84.0i)T^{2} \) |
| 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
−11.22819242297507965203414286390, −10.09449461299477188396007809538, −8.919557546610913389839758347622, −7.961745383540854470312348860964, −6.80678581254303469343764666386, −6.29308180297660061087948235533, −5.02988333187645558769237983351, −4.02492173299325131577347048145, −3.22681406730649817915255703169, −1.45636537875648754628031185346,
1.99931565030268570034526995467, 2.86120609177115460000424784791, 4.40052059134171200889795677528, 5.08776829281533608447679276266, 6.25861787399197360058533037052, 6.84823704997275280939585526241, 8.335886280459919997910100489845, 8.882592739483047862186445244473, 10.34881820576345003982387394275, 11.10499757382932968352117024594
