| L(s) = 1 | + (1.32 + 0.499i)2-s + (−2.67 + 0.867i)3-s + (1.50 + 1.32i)4-s + (2.16 − 0.573i)5-s + (−3.96 − 0.185i)6-s + (3.07 − 3.07i)7-s + (1.32 + 2.49i)8-s + (3.95 − 2.87i)9-s + (3.14 + 0.320i)10-s + (−2.14 + 0.340i)11-s + (−5.15 − 2.22i)12-s + (0.756 − 0.549i)13-s + (5.60 − 2.53i)14-s + (−5.27 + 3.40i)15-s + (0.505 + 3.96i)16-s + (2.82 + 5.53i)17-s + ⋯ |
| L(s) = 1 | + (0.935 + 0.353i)2-s + (−1.54 + 0.501i)3-s + (0.750 + 0.660i)4-s + (0.966 − 0.256i)5-s + (−1.61 − 0.0759i)6-s + (1.16 − 1.16i)7-s + (0.468 + 0.883i)8-s + (1.31 − 0.957i)9-s + (0.994 + 0.101i)10-s + (−0.647 + 0.102i)11-s + (−1.48 − 0.643i)12-s + (0.209 − 0.152i)13-s + (1.49 − 0.677i)14-s + (−1.36 + 0.880i)15-s + (0.126 + 0.991i)16-s + (0.684 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77940 + 0.706500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77940 + 0.706500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 - 0.499i)T \) |
| 5 | \( 1 + (-2.16 + 0.573i)T \) |
| good | 3 | \( 1 + (2.67 - 0.867i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-3.07 + 3.07i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.14 - 0.340i)T + (10.4 - 3.39i)T^{2} \) |
| 13 | \( 1 + (-0.756 + 0.549i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.82 - 5.53i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (2.30 + 4.53i)T + (-11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (-1.10 - 6.94i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.330 - 0.647i)T + (-17.0 - 23.4i)T^{2} \) |
| 31 | \( 1 + (-1.32 - 0.429i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.950 - 0.690i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.35 + 8.74i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + (-0.702 + 1.37i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-2.64 + 0.858i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-6.39 - 1.01i)T + (56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (6.44 - 1.02i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.0812 + 0.249i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.01 + 6.20i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.36 + 1.32i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (0.800 + 2.46i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-8.32 - 2.70i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (13.5 + 9.85i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.19 + 2.64i)T + (57.0 + 78.4i)T^{2} \) |
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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
−11.32403468315222384440081732463, −10.64067296575530655093572450810, −10.17104681054381927579339222201, −8.415000107735152110652266693107, −7.26600529742039790975527971985, −6.32977518358436786725211946293, −5.29471975405639596841802118323, −4.96907423854983692637158746147, −3.81946735920495652052007864924, −1.58045032495796921486143867047,
1.49798225504691032959389215293, 2.61928954042569740606405514980, 4.87301581737270932179996348176, 5.29754845882155685533222805362, 6.07699999697988573883110046808, 6.85728293352557133465045549624, 8.238313107665494934681711897497, 9.843150308618298848001871843550, 10.64401649187920026934848339027, 11.40040708911730763048594939785
