L(s) = 1 | + (−0.258 − 0.965i)2-s + (1.38 − 1.03i)3-s + (−0.866 + 0.499i)4-s + (−1.30 − 1.81i)5-s + (−1.36 − 1.06i)6-s + (−2.54 − 0.736i)7-s + (0.707 + 0.707i)8-s + (0.839 − 2.88i)9-s + (−1.41 + 1.73i)10-s + (−1.83 + 1.05i)11-s + (−0.680 + 1.59i)12-s + (3.28 − 3.28i)13-s + (−0.0535 + 2.64i)14-s + (−3.69 − 1.16i)15-s + (0.500 − 0.866i)16-s + (1.40 + 0.375i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (0.799 − 0.600i)3-s + (−0.433 + 0.249i)4-s + (−0.583 − 0.812i)5-s + (−0.556 − 0.436i)6-s + (−0.960 − 0.278i)7-s + (0.249 + 0.249i)8-s + (0.279 − 0.960i)9-s + (−0.447 + 0.547i)10-s + (−0.553 + 0.319i)11-s + (−0.196 + 0.459i)12-s + (0.910 − 0.910i)13-s + (−0.0143 + 0.706i)14-s + (−0.954 − 0.299i)15-s + (0.125 − 0.216i)16-s + (0.339 + 0.0910i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.439258 - 1.01616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.439258 - 1.01616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (-1.38 + 1.03i)T \) |
| 5 | \( 1 + (1.30 + 1.81i)T \) |
| 7 | \( 1 + (2.54 + 0.736i)T \) |
good | 11 | \( 1 + (1.83 - 1.05i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.28 + 3.28i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.40 - 0.375i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.72 - 2.72i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.50 + 1.47i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 7.20T + 29T^{2} \) |
| 31 | \( 1 + (0.528 + 0.914i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.22 + 1.66i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 1.86iT - 41T^{2} \) |
| 43 | \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.87 - 10.7i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.73 + 6.47i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.63 - 4.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.11 + 7.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.43 - 9.10i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (1.20 + 0.323i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (13.8 + 7.99i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.47 - 9.47i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.199 - 0.346i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.63 + 8.63i)T + 97iT^{2} \) |
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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
−12.30336448495060086839836536158, −11.11590046993313763755718500627, −9.864109727386165372355986115340, −9.103141374562743262550605988065, −8.065819561607694972911035555146, −7.33950959742021662510324548106, −5.65595544264747608386724815530, −3.91926539892677876581612802700, −3.00207739001638989596325957715, −1.01128283120737224521418567278,
2.93925928487732291480825331558, 3.90925934725723173802101213776, 5.49337906921627250819646578515, 6.84534386612174153445754334074, 7.65902206311826612946323702756, 8.854712582662406596403650177322, 9.528476969737731952661258442480, 10.61845844208957143732766294302, 11.53616727608653031365180324997, 13.16728020057920254133631639270