| L(s) = 1 | + (1.31 + 0.531i)2-s + (1.11 − 0.176i)3-s + (1.43 + 1.39i)4-s + (−2.17 + 0.507i)5-s + (1.55 + 0.361i)6-s + 4.59i·7-s + (1.13 + 2.58i)8-s + (−1.63 + 0.532i)9-s + (−3.12 − 0.493i)10-s + (5.03 − 2.56i)11-s + (1.84 + 1.30i)12-s + (−3.07 − 1.56i)13-s + (−2.44 + 6.01i)14-s + (−2.33 + 0.950i)15-s + (0.115 + 3.99i)16-s + (0.981 − 0.713i)17-s + ⋯ |
| L(s) = 1 | + (0.926 + 0.376i)2-s + (0.644 − 0.102i)3-s + (0.717 + 0.696i)4-s + (−0.973 + 0.226i)5-s + (0.635 + 0.147i)6-s + 1.73i·7-s + (0.402 + 0.915i)8-s + (−0.546 + 0.177i)9-s + (−0.987 − 0.155i)10-s + (1.51 − 0.773i)11-s + (0.533 + 0.375i)12-s + (−0.852 − 0.434i)13-s + (−0.652 + 1.60i)14-s + (−0.604 + 0.245i)15-s + (0.0287 + 0.999i)16-s + (0.238 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96371 + 1.47390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96371 + 1.47390i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.31 - 0.531i)T \) |
| 5 | \( 1 + (2.17 - 0.507i)T \) |
| good | 3 | \( 1 + (-1.11 + 0.176i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 - 4.59iT - 7T^{2} \) |
| 11 | \( 1 + (-5.03 + 2.56i)T + (6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (3.07 + 1.56i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.981 + 0.713i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.06 + 6.70i)T + (-18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (-1.41 - 0.459i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.83 + 1.08i)T + (27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (-1.40 + 1.01i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.06 + 4.04i)T + (-21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.56 + 0.507i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.39 - 1.39i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.49 + 4.72i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.580 - 3.66i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (6.12 - 12.0i)T + (-34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (0.782 + 1.53i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (-0.287 + 1.81i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-2.40 + 3.31i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (8.13 + 2.64i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (9.12 + 6.62i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.505 - 3.18i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.23 - 1.37i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.31 - 2.40i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.82994307799301816494285864466, −11.07231868307441571043636237797, −9.147277430657747289899993127493, −8.637398379486952644908816999564, −7.74112671055240802770569300875, −6.66101683167148484072510443596, −5.67400847445160709167270543054, −4.60066971433058217154080840751, −3.16003083279322691932368318355, −2.63560646745393777116939130976,
1.32702296552597898423410237110, 3.27007626648257747689158633285, 4.03024943787958162458273404211, 4.66467369222436977000637749322, 6.46967163372453065311958039984, 7.23878380961845812318635850304, 8.127644125074010654199089402330, 9.540161583573273070919849850486, 10.22098897088603511922778060370, 11.36815635664265738183720961833
