| L(s) = 1 | + (−0.615 + 0.710i)2-s + (0.841 − 0.540i)3-s + (0.158 + 1.10i)4-s + (−0.415 − 0.909i)5-s + (−0.133 + 0.930i)6-s + (0.864 − 0.253i)7-s + (−2.46 − 1.58i)8-s + (0.415 − 0.909i)9-s + (0.902 + 0.264i)10-s + (2.95 + 3.41i)11-s + (0.730 + 0.843i)12-s + (5.85 + 1.71i)13-s + (−0.351 + 0.770i)14-s + (−0.841 − 0.540i)15-s + (0.502 − 0.147i)16-s + (−0.180 + 1.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.435 + 0.502i)2-s + (0.485 − 0.312i)3-s + (0.0793 + 0.552i)4-s + (−0.185 − 0.406i)5-s + (−0.0546 + 0.379i)6-s + (0.326 − 0.0959i)7-s + (−0.871 − 0.560i)8-s + (0.138 − 0.303i)9-s + (0.285 + 0.0837i)10-s + (0.891 + 1.02i)11-s + (0.210 + 0.243i)12-s + (1.62 + 0.476i)13-s + (−0.0940 + 0.205i)14-s + (−0.217 − 0.139i)15-s + (0.125 − 0.0368i)16-s + (−0.0438 + 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19829 + 0.576493i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19829 + 0.576493i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (1.58 + 4.52i)T \) |
| good | 2 | \( 1 + (0.615 - 0.710i)T + (-0.284 - 1.97i)T^{2} \) |
| 7 | \( 1 + (-0.864 + 0.253i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.95 - 3.41i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.85 - 1.71i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.180 - 1.25i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.225 - 1.56i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.532 - 3.70i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.33 - 2.14i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (2.70 - 5.91i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.82 + 6.18i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.83 + 4.39i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 + (-10.5 + 3.09i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (11.9 + 3.52i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (7.38 + 4.74i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.50 + 1.73i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.72 - 1.98i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (2.41 + 16.8i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-1.81 - 0.534i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.771 - 1.68i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-6.62 + 4.25i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.28 + 7.20i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.94871576783553514124095031984, −10.67327663844954296245769405909, −9.346117868444204684824490056303, −8.676471290878488041787733064046, −8.036296134440886388857517721654, −6.96368831563456700999621122159, −6.25591857620703484286191906568, −4.41470118540408010506054902160, −3.48988313393870174431790844612, −1.62509227848374827889150034641,
1.25959066912948008350218266570, 2.88687887389845670954882364362, 3.96212301462985841283825022373, 5.61984740956062176326360300660, 6.40439689585143770668557738083, 7.945648496005710783441025626866, 8.788705604271007120436449952756, 9.495081092315727603156641811655, 10.57334796443431184103441291675, 11.23788653878089591421067625159
