| L(s) = 1 | + (−0.955 − 0.294i)2-s + (0.804 − 2.05i)3-s + (0.826 + 0.563i)4-s + (0.467 − 0.0705i)5-s + (−1.37 + 1.72i)6-s + (−0.623 − 0.781i)8-s + (−1.35 − 1.26i)9-s + (−0.467 − 0.0705i)10-s + (−0.346 + 0.321i)11-s + (1.82 − 1.24i)12-s + (0.0313 + 0.137i)13-s + (0.231 − 1.01i)15-s + (0.365 + 0.930i)16-s + (−0.505 − 6.74i)17-s + (0.926 + 1.60i)18-s + (2.32 − 4.02i)19-s + ⋯ |
| L(s) = 1 | + (−0.675 − 0.208i)2-s + (0.464 − 1.18i)3-s + (0.413 + 0.281i)4-s + (0.209 − 0.0315i)5-s + (−0.560 + 0.703i)6-s + (−0.220 − 0.276i)8-s + (−0.452 − 0.420i)9-s + (−0.147 − 0.0222i)10-s + (−0.104 + 0.0970i)11-s + (0.525 − 0.358i)12-s + (0.00869 + 0.0380i)13-s + (0.0598 − 0.262i)15-s + (0.0913 + 0.232i)16-s + (−0.122 − 1.63i)17-s + (0.218 + 0.378i)18-s + (0.533 − 0.923i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.559258 - 1.08194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.559258 - 1.08194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.955 + 0.294i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-0.804 + 2.05i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (-0.467 + 0.0705i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (0.346 - 0.321i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0313 - 0.137i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (0.505 + 6.74i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 4.12i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (4.42 + 2.12i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (0.190 + 0.330i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.92 + 6.08i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (-6.40 - 8.03i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (4.50 - 5.64i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (2.67 + 0.825i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (10.3 + 7.07i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-12.4 - 1.87i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (3.20 - 2.18i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-4.31 - 7.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.99 - 1.44i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (6.06 - 1.87i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (4.99 - 8.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.43 + 10.6i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-2.22 - 2.06i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
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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
−9.862055164184150298720081269474, −9.356785886598534083319510761544, −8.349421817088530049657116157633, −7.53518790735459649898766060340, −7.02733718117003081698305574340, −6.00146147018579361589509113638, −4.63329959860111766141088529818, −2.93582391664629103258693717025, −2.15711673933546425275490409722, −0.77385989581585008515504172221,
1.75723199361053572459011798664, 3.31774642060920892202286617688, 4.14611937730538924178038771619, 5.45845348969475117808837262712, 6.25540125513186227723018164501, 7.60396697776342918852537785940, 8.318131413759658866649974901653, 9.222125253497029211080008249014, 9.834385060227405833160142920974, 10.46312715591447277365574958107
