| L(s) = 1 | + (0.995 + 0.0950i)2-s + (0.690 − 0.723i)3-s + (0.981 + 0.189i)4-s + (−0.867 + 0.447i)5-s + (0.755 − 0.654i)6-s + (2.46 − 0.964i)7-s + (0.959 + 0.281i)8-s + (−0.0475 − 0.998i)9-s + (−0.905 + 0.362i)10-s + (0.472 + 4.95i)11-s + (0.814 − 0.580i)12-s + (2.98 + 0.428i)13-s + (2.54 − 0.726i)14-s + (−0.274 + 0.936i)15-s + (0.928 + 0.371i)16-s + (1.85 − 5.35i)17-s + ⋯ |
| L(s) = 1 | + (0.703 + 0.0672i)2-s + (0.398 − 0.417i)3-s + (0.490 + 0.0946i)4-s + (−0.387 + 0.199i)5-s + (0.308 − 0.267i)6-s + (0.931 − 0.364i)7-s + (0.339 + 0.0996i)8-s + (−0.0158 − 0.332i)9-s + (−0.286 + 0.114i)10-s + (0.142 + 1.49i)11-s + (0.235 − 0.167i)12-s + (0.827 + 0.118i)13-s + (0.679 − 0.194i)14-s + (−0.0709 + 0.241i)15-s + (0.232 + 0.0929i)16-s + (0.449 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.99302 - 0.170324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.99302 - 0.170324i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.995 - 0.0950i)T \) |
| 3 | \( 1 + (-0.690 + 0.723i)T \) |
| 7 | \( 1 + (-2.46 + 0.964i)T \) |
| 23 | \( 1 + (2.36 - 4.17i)T \) |
| good | 5 | \( 1 + (0.867 - 0.447i)T + (2.90 - 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.472 - 4.95i)T + (-10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-2.98 - 0.428i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 5.35i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (0.422 + 1.22i)T + (-14.9 + 11.7i)T^{2} \) |
| 29 | \( 1 + (4.26 + 4.92i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-5.19 + 1.25i)T + (27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (-11.2 + 0.535i)T + (36.8 - 3.51i)T^{2} \) |
| 41 | \( 1 + (1.72 - 2.68i)T + (-17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.98 + 6.76i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-7.03 - 4.06i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.83 + 8.68i)T + (-12.4 + 51.5i)T^{2} \) |
| 59 | \( 1 + (-0.920 - 2.29i)T + (-42.7 + 40.7i)T^{2} \) |
| 61 | \( 1 + (8.97 - 8.55i)T + (2.90 - 60.9i)T^{2} \) |
| 67 | \( 1 + (7.17 + 5.11i)T + (21.9 + 63.3i)T^{2} \) |
| 71 | \( 1 + (0.399 + 0.874i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.17 + 6.08i)T + (-67.7 - 27.1i)T^{2} \) |
| 79 | \( 1 + (6.61 - 8.40i)T + (-18.6 - 76.7i)T^{2} \) |
| 83 | \( 1 + (4.96 - 3.19i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (2.08 - 8.60i)T + (-79.1 - 40.7i)T^{2} \) |
| 97 | \( 1 + (-9.17 - 5.89i)T + (40.2 + 88.2i)T^{2} \) |
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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
−9.962537109676010078001483204695, −9.216912430014792535545963467581, −7.78642534257367809211852303253, −7.65494926543194358434346855045, −6.73115889079153722128381394440, −5.60416319508715611337590249826, −4.54175722594077106981894218292, −3.86765977525514244718748540609, −2.56291337922765189326734672975, −1.45034298172296944520081396856,
1.41328183436922670220484732934, 2.86213179914717745298552492012, 3.84867589246541304914964995518, 4.52624933643182991330971172967, 5.77406853170578512953612946988, 6.19169192729524354499550258164, 7.86872898694551832198869255395, 8.267059889775581790552138648982, 8.960824663414307893590934712618, 10.32805548800247120843604575554
