L(s) = 1 | + (1.59 − 2.75i)3-s + (−1.28 + 2.21i)5-s + (−3.56 − 6.16i)9-s − 4.96·11-s + (−2.28 − 3.95i)13-s + (4.07 + 7.05i)15-s + (2.28 − 3.95i)17-s + (0.697 + 4.30i)19-s + (0.893 + 1.54i)23-s + (−0.780 − 1.35i)25-s − 13.1·27-s + (−3.84 − 6.65i)29-s − 6.36·31-s + (−7.90 + 13.6i)33-s − 7.12·37-s + ⋯ |
L(s) = 1 | + (0.918 − 1.59i)3-s + (−0.572 + 0.992i)5-s + (−1.18 − 2.05i)9-s − 1.49·11-s + (−0.632 − 1.09i)13-s + (1.05 + 1.82i)15-s + (0.553 − 0.958i)17-s + (0.160 + 0.987i)19-s + (0.186 + 0.322i)23-s + (−0.156 − 0.270i)25-s − 2.52·27-s + (−0.713 − 1.23i)29-s − 1.14·31-s + (−1.37 + 2.38i)33-s − 1.17·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8439793560\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8439793560\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.697 - 4.30i)T \) |
good | 3 | \( 1 + (-1.59 + 2.75i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.28 - 2.21i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 + (2.28 + 3.95i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.28 + 3.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.893 - 1.54i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.84 + 6.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.36T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.28 + 3.96i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.07 - 7.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.28 - 3.95i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.59 - 2.75i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.28 + 5.68i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.98 + 5.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.893 - 1.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 13.2i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.07 + 7.05i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.39T + 83T^{2} \) |
| 89 | \( 1 + (0.842 + 1.45i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.62 + 4.54i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.143386129579345738019726065183, −7.982994588246555438495820634611, −7.52243385027961905794233054693, −7.40888274395982272699923971814, −6.13845137454861106688741241767, −5.28474433843060335443091484820, −3.40523515343781593190503374611, −2.95511754868336794445868667760, −2.01251757137446784241719311740, −0.29212685209668569000213088047,
2.15639424872154195928723125206, 3.29392176849403236236755819825, 4.13371225680253503070498848818, 4.97301981344827521216218300729, 5.34191187961267423810069103355, 7.17554049140179898393748176428, 8.039551381068635006336518359315, 8.746918059034118969648002242594, 9.170530683705467615700160592455, 10.09269433845270582164810546999