| L(s) = 1 | + (−0.104 − 0.994i)2-s + (−1.00 − 1.11i)3-s + (−0.978 + 0.207i)4-s + (2.10 + 0.763i)5-s + (−1.00 + 1.11i)6-s − 3.65·7-s + (0.309 + 0.951i)8-s + (0.0786 − 0.748i)9-s + (0.540 − 2.16i)10-s + (−0.0737 + 0.0535i)11-s + (1.21 + 0.881i)12-s + (0.453 − 4.31i)13-s + (0.382 + 3.63i)14-s + (−1.25 − 3.10i)15-s + (0.913 − 0.406i)16-s + (2.65 + 0.563i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (−0.579 − 0.643i)3-s + (−0.489 + 0.103i)4-s + (0.939 + 0.341i)5-s + (−0.409 + 0.454i)6-s − 1.38·7-s + (0.109 + 0.336i)8-s + (0.0262 − 0.249i)9-s + (0.170 − 0.686i)10-s + (−0.0222 + 0.0161i)11-s + (0.350 + 0.254i)12-s + (0.125 − 1.19i)13-s + (0.102 + 0.972i)14-s + (−0.324 − 0.802i)15-s + (0.228 − 0.101i)16-s + (0.643 + 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132902 + 0.245565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132902 + 0.245565i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (-2.10 - 0.763i)T \) |
| 19 | \( 1 + (3.99 - 1.75i)T \) |
| good | 3 | \( 1 + (1.00 + 1.11i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + (0.0737 - 0.0535i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.453 + 4.31i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-2.65 - 0.563i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (6.40 + 2.85i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (2.94 - 0.626i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.909 - 2.79i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.48 + 1.80i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (8.48 - 3.77i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.163 + 0.282i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0345 - 0.00734i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (11.5 - 2.44i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (4.08 - 1.81i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-4.78 - 2.12i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-3.80 + 4.22i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-5.92 - 6.58i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (0.842 + 8.01i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (2.33 + 2.58i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.978 - 3.01i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.55 + 2.47i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (5.87 + 6.52i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−9.894310366909815289215277272406, −8.877887898867489255099282297647, −7.80212802914639282486133178297, −6.59582321853547461052707999126, −6.15645568797084657998102751224, −5.36012770701580302926506811979, −3.70829060989079291858128427680, −2.90608431369596536884691138957, −1.62363106701617722212024954843, −0.13769925339844109422893167111,
2.01895349830822204440379356485, 3.66246426988590366196215296610, 4.63837827718172266453261840799, 5.56900206357456936706539555877, 6.23785396157020772232732053835, 6.87918103843269859783954943633, 8.145919367093038962173236257546, 9.160273248268468511689563734561, 9.778290130713014446230318044287, 10.15953416860545057395247443517
