| L(s) = 1 | + (1.36 − 0.386i)2-s + (−2.47 + 1.24i)3-s + (1.70 − 1.05i)4-s + (−0.190 − 0.428i)5-s + (−2.88 + 2.64i)6-s + (1.27 + 0.762i)7-s + (1.90 − 2.08i)8-s + (2.78 − 3.75i)9-s + (−0.424 − 0.510i)10-s + (0.698 + 0.545i)11-s + (−2.89 + 4.71i)12-s + (1.27 − 1.21i)13-s + (2.02 + 0.545i)14-s + (1.00 + 0.824i)15-s + (1.78 − 3.57i)16-s + (1.31 + 1.60i)17-s + ⋯ |
| L(s) = 1 | + (0.961 − 0.273i)2-s + (−1.42 + 0.718i)3-s + (0.850 − 0.525i)4-s + (−0.0850 − 0.191i)5-s + (−1.17 + 1.08i)6-s + (0.481 + 0.288i)7-s + (0.674 − 0.738i)8-s + (0.927 − 1.25i)9-s + (−0.134 − 0.161i)10-s + (0.210 + 0.164i)11-s + (−0.836 + 1.36i)12-s + (0.352 − 0.335i)13-s + (0.541 + 0.145i)14-s + (0.259 + 0.212i)15-s + (0.447 − 0.894i)16-s + (0.319 + 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81028 + 0.111322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81028 + 0.111322i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.386i)T \) |
| good | 3 | \( 1 + (2.47 - 1.24i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (0.190 + 0.428i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-1.27 - 0.762i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.698 - 0.545i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.27 + 1.21i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.31 - 1.60i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-1.06 - 6.11i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-7.55 - 3.57i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (0.456 + 0.258i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-2.12 + 0.423i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (2.41 + 1.53i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-4.24 - 4.68i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (0.107 + 0.324i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (11.0 + 5.90i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (2.41 - 1.36i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (5.95 - 6.25i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (0.547 + 7.41i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (1.78 + 1.54i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-5.48 - 0.813i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (7.46 - 4.47i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (0.296 + 0.0899i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (9.17 - 5.80i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-5.93 + 2.80i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-2.25 - 1.50i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.07972249032291024355687399012, −10.40168872715083242057701499649, −9.599357359828473445594688551494, −8.119338475214657800421682155293, −6.80070453711222361517756429071, −5.88272300248843161018768774106, −5.22941569145551503547228678479, −4.44974550556260374549739249340, −3.35409740392973644847324193938, −1.34518105074810424486697277080,
1.24626288351128149960702528333, 3.00439189764261128287634380399, 4.64800927778908645364720858627, 5.16974900791501449085422774815, 6.33210840650372110771499191277, 6.91332164046228910370848447125, 7.61505801814083990947740637951, 8.970764977024873756406311513003, 10.67311207085201646505458511260, 11.20599949541324323745674442233
