| L(s) = 1 | + (0.654 + 0.755i)2-s + (1.54 + 0.989i)3-s + (−0.142 + 0.989i)4-s + (−0.570 + 1.24i)5-s + (0.260 + 1.81i)6-s + (0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.146 + 0.320i)9-s + (−1.31 + 0.386i)10-s + (−0.889 + 1.02i)11-s + (−1.19 + 1.38i)12-s + (1.98 − 0.582i)13-s + (0.415 + 0.909i)14-s + (−2.11 + 1.35i)15-s + (−0.959 − 0.281i)16-s + (−0.380 − 2.64i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (0.889 + 0.571i)3-s + (−0.0711 + 0.494i)4-s + (−0.255 + 0.558i)5-s + (0.106 + 0.739i)6-s + (0.362 + 0.106i)7-s + (−0.297 + 0.191i)8-s + (0.0487 + 0.106i)9-s + (−0.416 + 0.122i)10-s + (−0.268 + 0.309i)11-s + (−0.346 + 0.399i)12-s + (0.550 − 0.161i)13-s + (0.111 + 0.243i)14-s + (−0.545 + 0.350i)15-s + (−0.239 − 0.0704i)16-s + (−0.0923 − 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0654 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41352 + 1.50934i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41352 + 1.50934i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (2.22 + 4.24i)T \) |
| good | 3 | \( 1 + (-1.54 - 0.989i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (0.570 - 1.24i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (0.889 - 1.02i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.98 + 0.582i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.380 + 2.64i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.297 - 2.06i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.204 - 1.42i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.10 + 1.99i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.235 - 0.515i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.37 + 7.37i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.31 - 0.845i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 + (2.46 + 0.724i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.55 + 0.456i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (5.20 - 3.34i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.32 - 4.99i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.38 - 3.90i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.50 + 10.4i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (6.74 - 1.98i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.319 - 0.698i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (8.57 + 5.51i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.53 + 14.3i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.92644228851618653980221785777, −10.91574214107981814730074512510, −9.911206850132079239376208767196, −8.851707043853812074255601502329, −8.108848592902826677099451900277, −7.12815447502583379185850349262, −5.98846014708701624799268256991, −4.66556641930910213337839974389, −3.65060117860343486731037412180, −2.61325891922718761942911950574,
1.44495181859295011733365134060, 2.77763154819111230320944882629, 4.03141908952439540138606686791, 5.18272273961279131278701324688, 6.48217275260898681914835681161, 7.86786785772344104380792294467, 8.434841053335176012018179544700, 9.406151420674816921866798786062, 10.67032569477848027632606995039, 11.45885906118989289659631115564
