| L(s) = 1 | + (−0.468 + 0.883i)2-s + (0.725 − 0.687i)3-s + (−0.561 − 0.827i)4-s + (−2.36 + 0.521i)5-s + (0.267 + 0.963i)6-s + (2.60 − 1.97i)7-s + (0.994 − 0.108i)8-s + (0.0541 − 0.998i)9-s + (0.648 − 2.33i)10-s + (−0.927 − 2.32i)11-s + (−0.976 − 0.214i)12-s + (−0.131 − 2.43i)13-s + (0.528 + 3.22i)14-s + (−1.36 + 2.00i)15-s + (−0.370 + 0.928i)16-s + (−0.651 − 0.494i)17-s + ⋯ |
| L(s) = 1 | + (−0.331 + 0.624i)2-s + (0.419 − 0.397i)3-s + (−0.280 − 0.413i)4-s + (−1.05 + 0.233i)5-s + (0.109 + 0.393i)6-s + (0.983 − 0.747i)7-s + (0.351 − 0.0382i)8-s + (0.0180 − 0.332i)9-s + (0.205 − 0.738i)10-s + (−0.279 − 0.701i)11-s + (−0.281 − 0.0620i)12-s + (−0.0365 − 0.674i)13-s + (0.141 + 0.862i)14-s + (−0.351 + 0.518i)15-s + (−0.0925 + 0.232i)16-s + (−0.157 − 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03583 - 0.359130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03583 - 0.359130i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.468 - 0.883i)T \) |
| 3 | \( 1 + (-0.725 + 0.687i)T \) |
| 59 | \( 1 + (7.48 - 1.70i)T \) |
| good | 5 | \( 1 + (2.36 - 0.521i)T + (4.53 - 2.09i)T^{2} \) |
| 7 | \( 1 + (-2.60 + 1.97i)T + (1.87 - 6.74i)T^{2} \) |
| 11 | \( 1 + (0.927 + 2.32i)T + (-7.98 + 7.56i)T^{2} \) |
| 13 | \( 1 + (0.131 + 2.43i)T + (-12.9 + 1.40i)T^{2} \) |
| 17 | \( 1 + (0.651 + 0.494i)T + (4.54 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-5.97 + 2.01i)T + (15.1 - 11.4i)T^{2} \) |
| 23 | \( 1 + (-4.88 + 2.93i)T + (10.7 - 20.3i)T^{2} \) |
| 29 | \( 1 + (0.464 + 0.876i)T + (-16.2 + 24.0i)T^{2} \) |
| 31 | \( 1 + (2.03 + 0.685i)T + (24.6 + 18.7i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 0.222i)T + (36.1 + 7.95i)T^{2} \) |
| 41 | \( 1 + (4.85 + 2.91i)T + (19.2 + 36.2i)T^{2} \) |
| 43 | \( 1 + (0.124 - 0.312i)T + (-31.2 - 29.5i)T^{2} \) |
| 47 | \( 1 + (-8.13 - 1.79i)T + (42.6 + 19.7i)T^{2} \) |
| 53 | \( 1 + (-2.07 - 7.46i)T + (-45.4 + 27.3i)T^{2} \) |
| 61 | \( 1 + (2.20 - 4.15i)T + (-34.2 - 50.4i)T^{2} \) |
| 67 | \( 1 + (9.99 - 1.08i)T + (65.4 - 14.4i)T^{2} \) |
| 71 | \( 1 + (-4.13 - 0.909i)T + (64.4 + 29.8i)T^{2} \) |
| 73 | \( 1 + (0.656 + 4.00i)T + (-69.1 + 23.3i)T^{2} \) |
| 79 | \( 1 + (-4.23 - 4.00i)T + (4.27 + 78.8i)T^{2} \) |
| 83 | \( 1 + (7.54 - 8.88i)T + (-13.4 - 81.9i)T^{2} \) |
| 89 | \( 1 + (-7.85 - 14.8i)T + (-49.9 + 73.6i)T^{2} \) |
| 97 | \( 1 + (-2.14 + 13.0i)T + (-91.9 - 30.9i)T^{2} \) |
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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
−11.19290498137527339443223185569, −10.62517632558971003309150617495, −9.219675169275542651262229538036, −8.245457113965771429877929359338, −7.61644295093293433501864766436, −7.06827398817360844870391342609, −5.54376740043793705494068058776, −4.37439696082104014232389443649, −3.08507060676786866737015009183, −0.886565614153099532588820540053,
1.78869012755121355235323750760, 3.26499276214665905023748396067, 4.42363089129400249757685063159, 5.26227689618370684618330931104, 7.29919858611073048787253489848, 7.986919952237947606193041865668, 8.862764868997826006566184439717, 9.604694052841988785425738087933, 10.77058719740003189501696848475, 11.69649020491524138568769213788
