| L(s) = 1 | + (−1.41 + 0.0734i)2-s + (−0.651 − 1.60i)3-s + (1.98 − 0.207i)4-s + (1.42 − 0.650i)5-s + (1.03 + 2.21i)6-s + (−2.49 − 2.61i)7-s + (−2.79 + 0.439i)8-s + (−2.15 + 2.09i)9-s + (−1.96 + 1.02i)10-s + (3.33 + 0.318i)11-s + (−1.62 − 3.05i)12-s + (−5.81 − 1.12i)13-s + (3.70 + 3.50i)14-s + (−1.97 − 1.86i)15-s + (3.91 − 0.825i)16-s + (−3.01 + 5.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 + 0.0519i)2-s + (−0.376 − 0.926i)3-s + (0.994 − 0.103i)4-s + (0.636 − 0.290i)5-s + (0.423 + 0.905i)6-s + (−0.941 − 0.987i)7-s + (−0.987 + 0.155i)8-s + (−0.717 + 0.696i)9-s + (−0.620 + 0.323i)10-s + (1.00 + 0.0959i)11-s + (−0.470 − 0.882i)12-s + (−1.61 − 0.311i)13-s + (0.991 + 0.937i)14-s + (−0.508 − 0.480i)15-s + (0.978 − 0.206i)16-s + (−0.731 + 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00165458 + 0.00262561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00165458 + 0.00262561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0734i)T \) |
| 3 | \( 1 + (0.651 + 1.60i)T \) |
| 67 | \( 1 + (8.14 - 0.806i)T \) |
| good | 5 | \( 1 + (-1.42 + 0.650i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (2.49 + 2.61i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-3.33 - 0.318i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (5.81 + 1.12i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (3.01 - 5.84i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (2.35 - 2.46i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.843 + 0.337i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (3.86 + 2.23i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.50 + 7.78i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (-3.98 - 6.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-11.3 + 0.538i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (1.72 + 2.67i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (4.66 + 3.66i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (0.847 - 1.31i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-6.99 - 8.07i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (5.18 - 0.494i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (11.2 - 5.82i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (5.34 - 0.510i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (3.47 - 1.20i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (5.27 - 7.40i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (-8.17 + 1.17i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-5.21 - 9.02i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.761159226080239598767973761268, −8.908076364114772925242211127279, −7.80113964039772711038425198849, −7.17483893668364616393599721619, −6.33781892801309757867406972841, −5.79964011005239772438255602750, −4.08866834864070602745425143104, −2.48440844925711791834008379222, −1.44768036587104046158442587928, −0.00212791938006953150850301521,
2.33007099077486625316604245632, 3.07289922021872639765967610431, 4.65172396738397977311759563241, 5.80452458841377504684523410936, 6.51506108274022734510222166359, 7.26497641756973018948256273883, 8.926515764551643259153125562417, 9.327043432626024788966890175502, 9.638340529536398220850832356196, 10.63239495398304530117546261618
