| L(s) = 1 | + (0.998 − 1.00i)2-s + (1.57 − 1.92i)3-s + (−0.00786 − 1.99i)4-s + (0.956 − 0.290i)5-s + (−0.352 − 3.50i)6-s + (1.57 + 0.312i)7-s + (−2.01 − 1.98i)8-s + (−0.624 − 3.13i)9-s + (0.664 − 1.24i)10-s + (−0.747 − 0.0736i)11-s + (−3.86 − 3.14i)12-s + (−1.27 + 4.20i)13-s + (1.88 − 1.26i)14-s + (0.952 − 2.30i)15-s + (−3.99 + 0.0314i)16-s + (0.465 + 1.12i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.708i)2-s + (0.912 − 1.11i)3-s + (−0.00393 − 0.999i)4-s + (0.427 − 0.129i)5-s + (−0.143 − 1.43i)6-s + (0.594 + 0.118i)7-s + (−0.711 − 0.702i)8-s + (−0.208 − 1.04i)9-s + (0.210 − 0.394i)10-s + (−0.225 − 0.0222i)11-s + (−1.11 − 0.907i)12-s + (−0.353 + 1.16i)13-s + (0.502 − 0.337i)14-s + (0.246 − 0.593i)15-s + (−0.999 + 0.00786i)16-s + (0.112 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39439 - 2.72337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39439 - 2.72337i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 + 1.00i)T \) |
| 5 | \( 1 + (-0.956 + 0.290i)T \) |
| good | 3 | \( 1 + (-1.57 + 1.92i)T + (-0.585 - 2.94i)T^{2} \) |
| 7 | \( 1 + (-1.57 - 0.312i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.747 + 0.0736i)T + (10.7 + 2.14i)T^{2} \) |
| 13 | \( 1 + (1.27 - 4.20i)T + (-10.8 - 7.22i)T^{2} \) |
| 17 | \( 1 + (-0.465 - 1.12i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-2.40 + 1.28i)T + (10.5 - 15.7i)T^{2} \) |
| 23 | \( 1 + (2.14 - 3.21i)T + (-8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.496 - 5.03i)T + (-28.4 + 5.65i)T^{2} \) |
| 31 | \( 1 + (0.858 + 0.858i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.0179 - 0.0335i)T + (-20.5 - 30.7i)T^{2} \) |
| 41 | \( 1 + (-0.0467 - 0.0312i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (3.70 + 4.51i)T + (-8.38 + 42.1i)T^{2} \) |
| 47 | \( 1 + (-6.11 + 2.53i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.251 - 2.55i)T + (-51.9 - 10.3i)T^{2} \) |
| 59 | \( 1 + (-2.01 - 6.62i)T + (-49.0 + 32.7i)T^{2} \) |
| 61 | \( 1 + (10.4 + 8.55i)T + (11.9 + 59.8i)T^{2} \) |
| 67 | \( 1 + (2.11 + 1.73i)T + (13.0 + 65.7i)T^{2} \) |
| 71 | \( 1 + (1.41 - 7.09i)T + (-65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (7.51 - 1.49i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-13.4 - 5.56i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (7.06 + 13.2i)T + (-46.1 + 69.0i)T^{2} \) |
| 89 | \( 1 + (1.08 + 1.61i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-5.02 - 5.02i)T + 97iT^{2} \) |
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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
−10.34125307972243987458095689244, −9.326506878461842367348091756399, −8.676578093227370548834057711359, −7.51994375753691994039690983710, −6.74296149034056433015464896153, −5.62476753950928617889773525442, −4.60206829332315426265653448732, −3.28457266575985090095734733386, −2.15829040195882228338617724341, −1.47672841466859660760588267356,
2.53673067741494543723259348135, 3.38451373684007957875931220892, 4.47593115452331427609109601216, 5.19584599019060110711043777071, 6.19845603832404401671796473667, 7.60615074213990405820601309686, 8.112309780270616999359255933392, 9.051564796887419534200589220886, 9.933662567173035515862647402378, 10.66665564353815036674190341876
