| L(s) = 1 | + (1.99 + 0.134i)2-s + (−1.35 − 1.07i)3-s + (3.96 + 0.536i)4-s + (5.34 + 2.57i)5-s + (−2.55 − 2.33i)6-s + (7.67 + 6.12i)7-s + (7.83 + 1.60i)8-s + (0.667 + 2.92i)9-s + (10.3 + 5.85i)10-s + (−20.1 − 4.59i)11-s + (−4.78 − 5.00i)12-s + (−3.27 + 14.3i)13-s + (14.4 + 13.2i)14-s + (−4.45 − 9.25i)15-s + (15.4 + 4.24i)16-s − 2.90·17-s + ⋯ |
| L(s) = 1 | + (0.997 + 0.0671i)2-s + (−0.451 − 0.359i)3-s + (0.990 + 0.134i)4-s + (1.06 + 0.514i)5-s + (−0.426 − 0.389i)6-s + (1.09 + 0.874i)7-s + (0.979 + 0.200i)8-s + (0.0741 + 0.324i)9-s + (1.03 + 0.585i)10-s + (−1.82 − 0.417i)11-s + (−0.399 − 0.417i)12-s + (−0.252 + 1.10i)13-s + (1.03 + 0.946i)14-s + (−0.297 − 0.617i)15-s + (0.964 + 0.265i)16-s − 0.170·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.875 - 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.23274 + 0.832281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.23274 + 0.832281i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 - 0.134i)T \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 29 | \( 1 + (-20.8 + 20.1i)T \) |
| good | 5 | \( 1 + (-5.34 - 2.57i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-7.67 - 6.12i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (20.1 + 4.59i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (3.27 - 14.3i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 2.90T + 289T^{2} \) |
| 19 | \( 1 + (-13.6 + 10.8i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (2.20 + 4.57i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (5.95 - 12.3i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (12.8 + 56.4i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 - 65.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-9.17 - 19.0i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (29.1 + 6.65i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (63.1 + 30.4i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 98.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (6.90 - 8.66i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (3.08 - 0.703i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-7.00 - 1.59i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-63.1 + 30.3i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (110. - 25.2i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (83.7 - 66.7i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (75.8 + 36.5i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-15.5 - 19.5i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−11.29127797022336415709704107151, −10.91375612431224670488716311546, −9.733913204308322052722371248493, −8.268718732501569983067592784371, −7.31689664045328584628738949653, −6.19801860478581023496248981689, −5.44428979372258864679129117508, −4.75137145605227898316881024602, −2.65507276228045535926891159846, −2.00327922426872955803091002032,
1.36678575800536106710432347743, 2.84186940380636815723651429303, 4.53234255342667705902215578149, 5.20651223121235213139211485726, 5.80189187592554373367034213555, 7.36251376327614683234822947100, 8.070020996098793675936710418436, 9.890283496492938899159936664705, 10.42049691712430469335018488812, 11.09994455937027061405784193981
