| 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
−11.37653319913835315104578124075, −10.17773193766092057615874262530, −9.812694385543168169325962215966, −8.739043264660979422865605171664, −8.011808143957370955078440273940, −6.13556934561178084707198217536, −5.30210357391810541250951189527, −4.31363540690442787666139978769, −2.63958605054189789194164735017, −1.15882845081274545428313625862,
1.94717062095311172984498385419, 3.58154150423671895223559181019, 5.11405637229607429007718677295, 6.37552469083110810305346944089, 6.88024966587353405666360606290, 7.60080108741014223863374724910, 9.474180519705741040495659269282, 9.732458325010728299355247497957, 10.58259739374409231194020336589, 12.02126357558361557043952596730
