| L(s) = 1 | + (1.40 + 0.151i)2-s + (−1.82 − 0.179i)3-s + (1.95 + 0.426i)4-s + (0.471 + 0.881i)5-s + (−2.53 − 0.528i)6-s + (0.716 + 3.60i)7-s + (2.68 + 0.896i)8-s + (0.340 + 0.0678i)9-s + (0.528 + 1.31i)10-s + (−2.48 − 2.03i)11-s + (−3.48 − 1.12i)12-s + (2.57 + 1.37i)13-s + (0.460 + 5.17i)14-s + (−0.700 − 1.69i)15-s + (3.63 + 1.66i)16-s + (−1.64 + 3.96i)17-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.107i)2-s + (−1.05 − 0.103i)3-s + (0.976 + 0.213i)4-s + (0.210 + 0.394i)5-s + (−1.03 − 0.215i)6-s + (0.270 + 1.36i)7-s + (0.948 + 0.317i)8-s + (0.113 + 0.0226i)9-s + (0.167 + 0.414i)10-s + (−0.749 − 0.614i)11-s + (−1.00 − 0.325i)12-s + (0.714 + 0.381i)13-s + (0.123 + 1.38i)14-s + (−0.180 − 0.436i)15-s + (0.908 + 0.417i)16-s + (−0.398 + 0.962i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41353 + 1.26002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41353 + 1.26002i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.151i)T \) |
| 5 | \( 1 + (-0.471 - 0.881i)T \) |
| good | 3 | \( 1 + (1.82 + 0.179i)T + (2.94 + 0.585i)T^{2} \) |
| 7 | \( 1 + (-0.716 - 3.60i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (2.48 + 2.03i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.57 - 1.37i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (1.64 - 3.96i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (4.99 - 1.51i)T + (15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (-0.267 + 0.178i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-5.07 - 6.18i)T + (-5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-0.257 + 0.257i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.93 + 9.66i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (-4.60 - 6.88i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (6.84 - 0.674i)T + (42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-6.78 - 2.81i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.97 + 2.40i)T + (-10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (0.227 - 0.121i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (-1.03 + 10.5i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 10.9i)T + (-65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (3.62 - 0.721i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.47 + 7.41i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (14.5 - 6.01i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (0.861 + 2.83i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (-0.378 - 0.252i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-9.07 + 9.07i)T - 97iT^{2} \) |
| 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
−10.98731545522357717080109898573, −10.50359276610441072604382584885, −8.827494521098955856611161290616, −8.119664902953396326407062346992, −6.65341560429385616357542176129, −6.03155861290732104027656327234, −5.57644262772601004947243416277, −4.52139947276757386904926011759, −3.10531666293260994048207305483, −1.97201309867085432974645490335,
0.854456231003206073747307914691, 2.58525368619290097057856772346, 4.21904104702632139765629822501, 4.73230737908043953039034291994, 5.66652222213287705545656458799, 6.57871690255292442850904364001, 7.37107202108886225881551179078, 8.463493484424936085478787316545, 10.20083082015479798808716798359, 10.43512175228856157757511818021
