| L(s) = 1 | + (1.22 + 0.701i)2-s + (−2.69 + 0.304i)3-s + (1.01 + 1.72i)4-s + (−1.20 + 2.49i)5-s + (−3.52 − 1.51i)6-s + (0.625 + 0.498i)7-s + (0.0419 + 2.82i)8-s + (4.27 − 0.974i)9-s + (−3.22 + 2.22i)10-s + (4.00 − 2.51i)11-s + (−3.26 − 4.33i)12-s + (−2.59 − 0.593i)13-s + (0.418 + 1.05i)14-s + (2.48 − 7.09i)15-s + (−1.93 + 3.50i)16-s + (4.12 − 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.868 + 0.495i)2-s + (−1.55 + 0.175i)3-s + (0.508 + 0.861i)4-s + (−0.537 + 1.11i)5-s + (−1.44 − 0.620i)6-s + (0.236 + 0.188i)7-s + (0.0148 + 0.999i)8-s + (1.42 − 0.324i)9-s + (−1.01 + 0.702i)10-s + (1.20 − 0.757i)11-s + (−0.943 − 1.25i)12-s + (−0.720 − 0.164i)13-s + (0.111 + 0.281i)14-s + (0.641 − 1.83i)15-s + (−0.482 + 0.875i)16-s + (0.999 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.622168 + 0.811376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.622168 + 0.811376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.701i)T \) |
| 29 | \( 1 + (-5.26 + 1.11i)T \) |
| good | 3 | \( 1 + (2.69 - 0.304i)T + (2.92 - 0.667i)T^{2} \) |
| 5 | \( 1 + (1.20 - 2.49i)T + (-3.11 - 3.90i)T^{2} \) |
| 7 | \( 1 + (-0.625 - 0.498i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-4.00 + 2.51i)T + (4.77 - 9.91i)T^{2} \) |
| 13 | \( 1 + (2.59 + 0.593i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.12 + 4.12i)T - 17iT^{2} \) |
| 19 | \( 1 + (0.647 - 5.74i)T + (-18.5 - 4.22i)T^{2} \) |
| 23 | \( 1 + (1.26 + 2.62i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (0.0981 + 0.280i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (1.42 - 2.27i)T + (-16.0 - 33.3i)T^{2} \) |
| 41 | \( 1 + (-6.66 - 6.66i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.02 + 0.357i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (1.28 + 2.03i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (12.1 + 5.87i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 + (0.603 + 5.35i)T + (-59.4 + 13.5i)T^{2} \) |
| 67 | \( 1 + (-1.16 - 5.10i)T + (-60.3 + 29.0i)T^{2} \) |
| 71 | \( 1 + (1.67 - 7.31i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (3.02 + 1.05i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-4.66 + 7.42i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (-7.68 + 6.13i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (1.83 - 0.641i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (0.198 - 1.75i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−14.26041297001171127136728977030, −12.46296265935835899981622368367, −11.74104832130750908986644336405, −11.26359994286153971736812783225, −10.06899592400637404240729839942, −7.962531649249400781769343891339, −6.75706759287984815682178759637, −6.05048796987167520448717449927, −4.83047172703048842229430496239, −3.39837251111545633725365410661,
1.20342902098964184588748930687, 4.26530315459281268763933166286, 4.95017994113268478520913394718, 6.16602325942598006306019743287, 7.33026410619435051931590745540, 9.307213741162741938957608217327, 10.56223462017511877459190738078, 11.56388273480638456033825334279, 12.35396895111538154447320189370, 12.56308170508206744572573179322
