| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.182 − 0.400i)3-s + (−0.959 − 0.281i)4-s + (2.48 + 2.87i)5-s + (0.421 − 0.123i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (1.83 − 2.12i)9-s + (−3.19 + 2.05i)10-s + (0.610 + 4.24i)11-s + (0.0625 + 0.435i)12-s + (−2.14 + 1.38i)13-s + (0.654 − 0.755i)14-s + (0.694 − 1.52i)15-s + (0.841 + 0.540i)16-s + (3.58 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.105 − 0.230i)3-s + (−0.479 − 0.140i)4-s + (1.11 + 1.28i)5-s + (0.172 − 0.0505i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (0.612 − 0.707i)9-s + (−1.01 + 0.650i)10-s + (0.183 + 1.27i)11-s + (0.0180 + 0.125i)12-s + (−0.596 + 0.383i)13-s + (0.175 − 0.201i)14-s + (0.179 − 0.392i)15-s + (0.210 + 0.135i)16-s + (0.868 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0356 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0356 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.971064 + 0.937050i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.971064 + 0.937050i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (1.68 - 4.49i)T \) |
| good | 3 | \( 1 + (0.182 + 0.400i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-2.48 - 2.87i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.610 - 4.24i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.14 - 1.38i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.58 + 1.05i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (1.18 + 0.348i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-5.13 + 1.50i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.97 - 6.51i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-5.05 + 5.83i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.147 - 0.170i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.98 + 8.72i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.240T + 47T^{2} \) |
| 53 | \( 1 + (8.72 + 5.60i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.17 + 3.97i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.72 + 8.14i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.83 + 12.7i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.00 + 6.95i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.22 - 0.653i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.54 + 2.92i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (1.65 - 1.91i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (1.27 + 2.79i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-8.97 - 10.3i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−12.02126357558361557043952596730, −10.58259739374409231194020336589, −9.732458325010728299355247497957, −9.474180519705741040495659269282, −7.60080108741014223863374724910, −6.88024966587353405666360606290, −6.37552469083110810305346944089, −5.11405637229607429007718677295, −3.58154150423671895223559181019, −1.94717062095311172984498385419,
1.15882845081274545428313625862, 2.63958605054189789194164735017, 4.31363540690442787666139978769, 5.30210357391810541250951189527, 6.13556934561178084707198217536, 8.011808143957370955078440273940, 8.739043264660979422865605171664, 9.812694385543168169325962215966, 10.17773193766092057615874262530, 11.37653319913835315104578124075
