| L(s) = 1 | + (−0.586 − 1.28i)2-s + (−0.342 + 0.939i)3-s + (−1.31 + 1.50i)4-s + (−0.951 + 0.167i)5-s + (1.40 − 0.110i)6-s + (1.03 − 1.79i)7-s + (2.71 + 0.803i)8-s + (−0.766 − 0.642i)9-s + (0.773 + 1.12i)10-s + (−3.62 + 2.09i)11-s + (−0.969 − 1.74i)12-s + (0.843 + 2.31i)13-s + (−2.92 − 0.281i)14-s + (0.167 − 0.951i)15-s + (−0.556 − 3.96i)16-s + (−2.32 + 1.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.414 − 0.909i)2-s + (−0.197 + 0.542i)3-s + (−0.656 + 0.754i)4-s + (−0.425 + 0.0750i)5-s + (0.575 − 0.0453i)6-s + (0.392 − 0.680i)7-s + (0.958 + 0.283i)8-s + (−0.255 − 0.214i)9-s + (0.244 + 0.355i)10-s + (−1.09 + 0.631i)11-s + (−0.279 − 0.504i)12-s + (0.233 + 0.642i)13-s + (−0.781 − 0.0752i)14-s + (0.0433 − 0.245i)15-s + (−0.139 − 0.990i)16-s + (−0.563 + 0.472i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.232192 + 0.294880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.232192 + 0.294880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.586 + 1.28i)T \) |
| 3 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (1.55 + 4.07i)T \) |
| good | 5 | \( 1 + (0.951 - 0.167i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-1.03 + 1.79i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.62 - 2.09i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.843 - 2.31i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.32 - 1.94i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.01 - 5.73i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.86 - 6.99i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.61 - 2.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.58iT - 37T^{2} \) |
| 41 | \( 1 + (-0.482 - 0.175i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.99 + 0.351i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.73 + 4.81i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.64 - 0.641i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (5.90 + 7.03i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-12.7 - 2.25i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.45 + 5.30i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.886 + 5.02i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (9.35 + 3.40i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (9.58 + 3.48i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.20 + 0.696i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.0 + 4.01i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (10.7 - 9.03i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.10969556275671196448064877514, −10.63121220882684500392000857963, −9.698972135040662138052229471992, −8.846219061791748042217086073046, −7.82059973392197741875729099926, −7.03781061590742547094905144711, −5.21538795838509388911231059027, −4.34510897474470389537474165861, −3.42524815517588382859874466902, −1.85949598728399510873242578545,
0.26559580707348156629537586376, 2.29092445863777032619575478816, 4.20928665387341411318512962804, 5.54844540135779494357027677494, 5.97590289972009111879941368691, 7.31170258746423791503742120281, 8.143364902588058935632252549890, 8.510379377148447553386036849179, 9.812554651924187603742710599614, 10.78730878511623157285895107172
