| 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
−13.67054734612486278245759631093, −12.50681959446754490911425426380, −11.15170220945804245481366710028, −10.62318872185450700687083003042, −9.356086961403654201401485492585, −7.918926406802593919489759668037, −7.39502432684241840107864604381, −6.10550576666165099223073715562, −3.17169615723551220607539204614, −2.07679307112854286513382684967,
1.98011337309793261168078332057, 4.83882890940067136129357717247, 5.44053857691343158726900942633, 7.73128402120173863598677556543, 8.512255318858626645158433087544, 9.336647122995862947482647259759, 10.20404900944762191955159022898, 11.40962969486043635218292647780, 12.85286826019142272885201391891, 14.06870118970922203672942966434
