| L(s) = 1 | + (0.678 + 1.24i)2-s + (−1.44 − 0.388i)3-s + (−1.08 + 1.68i)4-s + (−1.54 − 1.62i)5-s + (−0.500 − 2.06i)6-s + (0.445 − 2.60i)7-s + (−2.82 − 0.200i)8-s + (−0.649 − 0.374i)9-s + (0.965 − 3.01i)10-s + (1.03 + 1.79i)11-s + (2.21 − 2.01i)12-s + (−3.78 − 3.78i)13-s + (3.53 − 1.21i)14-s + (1.60 + 2.94i)15-s + (−1.66 − 3.63i)16-s + (0.443 − 1.65i)17-s + ⋯ |
| L(s) = 1 | + (0.479 + 0.877i)2-s + (−0.836 − 0.224i)3-s + (−0.540 + 0.841i)4-s + (−0.689 − 0.724i)5-s + (−0.204 − 0.841i)6-s + (0.168 − 0.985i)7-s + (−0.997 − 0.0707i)8-s + (−0.216 − 0.124i)9-s + (0.305 − 0.952i)10-s + (0.312 + 0.541i)11-s + (0.640 − 0.582i)12-s + (−1.04 − 1.04i)13-s + (0.945 − 0.324i)14-s + (0.414 + 0.760i)15-s + (−0.416 − 0.909i)16-s + (0.107 − 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0901 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0901 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.395510 - 0.361312i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395510 - 0.361312i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.678 - 1.24i)T \) |
| 5 | \( 1 + (1.54 + 1.62i)T \) |
| 7 | \( 1 + (-0.445 + 2.60i)T \) |
| good | 3 | \( 1 + (1.44 + 0.388i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.03 - 1.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.78 + 3.78i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.443 + 1.65i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.69 + 1.79i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 8.01T + 29T^{2} \) |
| 31 | \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.02 + 3.81i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 1.76T + 41T^{2} \) |
| 43 | \( 1 + (7.50 - 7.50i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.286 + 1.06i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.20 + 8.24i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-8.33 + 4.81i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.39 - 4.27i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 3.23i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 5.34iT - 71T^{2} \) |
| 73 | \( 1 + (-0.565 - 0.151i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.44 + 5.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.12 + 9.12i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.29 - 3.63i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.7 + 10.7i)T + 97iT^{2} \) |
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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
−11.82795995187354473525038500781, −11.00854371587491604168846877209, −9.591998768221555080917514793705, −8.471527639972972204395706382232, −7.40096164713663479728005415800, −6.86799906540037723426665433957, −5.34511132590331969573309328647, −4.79696683242546175720987152223, −3.49802916064618602239757490556, −0.37442716974290217167060852070,
2.28874539268110666541765927948, 3.63483429684153077478664064868, 4.91331835696230901144087232029, 5.79263937629656139183651601480, 6.88767679215065889903605506366, 8.496462647486905430105469594243, 9.470743746571479048329010354001, 10.63826278820674180063893879860, 11.38560076641660957964763043761, 11.76098451712301699168241780436
