| L(s) = 1 | + (−1.27 + 0.607i)2-s + (0.971 + 1.43i)3-s + (1.26 − 1.55i)4-s + (1.29 − 3.56i)5-s + (−2.11 − 1.24i)6-s + (2.58 + 0.455i)7-s + (−0.671 + 2.74i)8-s + (−1.11 + 2.78i)9-s + (0.507 + 5.33i)10-s + (−3.39 + 1.23i)11-s + (3.45 + 0.303i)12-s + (−0.819 + 0.688i)13-s + (−3.57 + 0.985i)14-s + (6.37 − 1.60i)15-s + (−0.811 − 3.91i)16-s + (0.980 − 0.566i)17-s + ⋯ |
| L(s) = 1 | + (−0.903 + 0.429i)2-s + (0.561 + 0.827i)3-s + (0.631 − 0.775i)4-s + (0.580 − 1.59i)5-s + (−0.862 − 0.506i)6-s + (0.975 + 0.171i)7-s + (−0.237 + 0.971i)8-s + (−0.370 + 0.928i)9-s + (0.160 + 1.68i)10-s + (−1.02 + 0.372i)11-s + (0.996 + 0.0875i)12-s + (−0.227 + 0.190i)13-s + (−0.954 + 0.263i)14-s + (1.64 − 0.413i)15-s + (−0.202 − 0.979i)16-s + (0.237 − 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.878776 + 0.234586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.878776 + 0.234586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 - 0.607i)T \) |
| 3 | \( 1 + (-0.971 - 1.43i)T \) |
| good | 5 | \( 1 + (-1.29 + 3.56i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.58 - 0.455i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.39 - 1.23i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.819 - 0.688i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.980 + 0.566i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0627 + 0.0362i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0731 - 0.414i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.192 - 0.229i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (6.88 - 1.21i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.377 - 0.654i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.36 - 7.58i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.18 + 8.74i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0443 - 0.251i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 12.0iT - 53T^{2} \) |
| 59 | \( 1 + (11.7 + 4.27i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.13 - 6.43i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.37 - 5.21i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.35 - 11.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.578 - 1.00i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.09 + 6.07i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.00 - 2.51i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.07 - 1.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.81 + 0.660i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.06870118970922203672942966434, −12.85286826019142272885201391891, −11.40962969486043635218292647780, −10.20404900944762191955159022898, −9.336647122995862947482647259759, −8.512255318858626645158433087544, −7.73128402120173863598677556543, −5.44053857691343158726900942633, −4.83882890940067136129357717247, −1.98011337309793261168078332057,
2.07679307112854286513382684967, 3.17169615723551220607539204614, 6.10550576666165099223073715562, 7.39502432684241840107864604381, 7.918926406802593919489759668037, 9.356086961403654201401485492585, 10.62318872185450700687083003042, 11.15170220945804245481366710028, 12.50681959446754490911425426380, 13.67054734612486278245759631093
