| L(s) = 1 | + (−0.594 − 1.28i)2-s + (0.300 − 0.357i)3-s + (−1.29 + 1.52i)4-s + (0.218 + 2.22i)5-s + (−0.637 − 0.172i)6-s + (−1.68 − 2.91i)7-s + (2.72 + 0.753i)8-s + (0.483 + 2.73i)9-s + (2.72 − 1.60i)10-s + (0.107 + 0.0619i)11-s + (0.157 + 0.921i)12-s + (4.63 − 3.89i)13-s + (−2.73 + 3.88i)14-s + (0.862 + 0.590i)15-s + (−0.653 − 3.94i)16-s + (5.27 + 0.929i)17-s + ⋯ |
| L(s) = 1 | + (−0.420 − 0.907i)2-s + (0.173 − 0.206i)3-s + (−0.646 + 0.762i)4-s + (0.0976 + 0.995i)5-s + (−0.260 − 0.0705i)6-s + (−0.635 − 1.10i)7-s + (0.963 + 0.266i)8-s + (0.161 + 0.913i)9-s + (0.862 − 0.506i)10-s + (0.0323 + 0.0186i)11-s + (0.0454 + 0.265i)12-s + (1.28 − 1.07i)13-s + (−0.731 + 1.03i)14-s + (0.222 + 0.152i)15-s + (−0.163 − 0.986i)16-s + (1.27 + 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06430 - 0.435576i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06430 - 0.435576i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.594 + 1.28i)T \) |
| 5 | \( 1 + (-0.218 - 2.22i)T \) |
| 19 | \( 1 + (-4.35 + 0.115i)T \) |
| good | 3 | \( 1 + (-0.300 + 0.357i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (1.68 + 2.91i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.107 - 0.0619i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.63 + 3.89i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.27 - 0.929i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.96 - 1.07i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.07 + 0.542i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.24 - 5.61i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + (-1.73 + 2.06i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (11.3 - 4.11i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.96 + 11.1i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.110 + 0.0402i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.756 - 4.28i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.90 + 2.51i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.55 + 0.274i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.80 + 1.02i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (4.12 - 4.91i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.67 - 7.27i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.57 - 6.18i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.78 + 3.31i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.90 + 10.8i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.92300541291354921916360938582, −10.35391726140690813448875388138, −9.911413867584367138577394738682, −8.396586843694881644810224588544, −7.64449093561612276516948144826, −6.80550417730644069795714865383, −5.26562915781494936493825102898, −3.56997432378554957390099405632, −3.08706321005333993979025446873, −1.26362748904523026849814128077,
1.21114404155438846868055188103, 3.49750794801051090172262970775, 4.82356883759140428553508555936, 5.88478302481250768330333488845, 6.52474411119404561679987131582, 7.938746512696892632660820375900, 8.934592183706420245633127282198, 9.250722269388112547379560458513, 10.02470282027327908918527013552, 11.63830483270518802634975369761
