| L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.415 − 0.909i)6-s + (−0.730 − 0.843i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)10-s + (−1.85 − 1.19i)11-s + (0.841 + 0.540i)12-s + (2.37 − 2.74i)13-s + (1.07 + 0.314i)14-s + (−0.142 − 0.989i)15-s + (−0.654 − 0.755i)16-s + (0.952 + 2.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.429 + 0.125i)5-s + (−0.169 − 0.371i)6-s + (−0.276 − 0.318i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.207 − 0.238i)10-s + (−0.558 − 0.359i)11-s + (0.242 + 0.156i)12-s + (0.660 − 0.761i)13-s + (0.286 + 0.0839i)14-s + (−0.0367 − 0.255i)15-s + (−0.163 − 0.188i)16-s + (0.230 + 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849489 - 0.0938465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849489 - 0.0938465i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.43 + 1.83i)T \) |
| good | 7 | \( 1 + (0.730 + 0.843i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.85 + 1.19i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.37 + 2.74i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.952 - 2.08i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.49 + 3.27i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.780 + 1.70i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.420 + 2.92i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-6.83 - 2.00i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (2.96 - 0.871i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.768 + 5.34i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 + (-2.36 - 2.72i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.27 + 4.93i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.549 - 3.82i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.00 + 3.21i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (10.6 - 6.83i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.228 - 0.500i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-7.38 + 8.51i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-14.7 - 4.33i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.241 + 1.67i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (1.17 - 0.345i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.49390632493709571371293616745, −9.578757323728813851217315189601, −8.655231271669229057809131839643, −7.960262070673588941994787677628, −7.02772164673835747376062347481, −6.00097819986036139820223662061, −5.11568709936533313286086986345, −3.87314128621586387831699212362, −2.79450825429289727159130143950, −0.65136900713605599308496352004,
1.20851833129771555749627151838, 2.57788719593806889105072019581, 3.70878341579955755847884962258, 5.06893919203947534483266606579, 6.23100797630246398465082541594, 7.22433189250975228953409345642, 7.87834957660090567165088281511, 8.851418172678770256453585177980, 9.507706264882383445850740438060, 10.57962042963195095818521024847
