| L(s) = 1 | + (−0.951 + 0.309i)2-s + (1.11 − 1.32i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (−0.654 + 1.60i)6-s + (0.396 + 3.76i)7-s + (−0.587 + 0.809i)8-s + (−0.499 − 2.95i)9-s + (−0.669 + 0.743i)10-s + (−3.50 − 1.55i)11-s + (0.127 − 1.72i)12-s + (1.20 + 5.67i)13-s + (−1.54 − 3.46i)14-s + (0.307 − 1.70i)15-s + (0.309 − 0.951i)16-s + (2.94 − 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.645 − 0.763i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (−0.267 + 0.654i)6-s + (0.149 + 1.42i)7-s + (−0.207 + 0.286i)8-s + (−0.166 − 0.986i)9-s + (−0.211 + 0.235i)10-s + (−1.05 − 0.470i)11-s + (0.0367 − 0.498i)12-s + (0.334 + 1.57i)13-s + (−0.412 − 0.925i)14-s + (0.0792 − 0.440i)15-s + (0.0772 − 0.237i)16-s + (0.714 − 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55741 + 0.142476i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55741 + 0.142476i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-1.11 + 1.32i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-0.212 + 5.56i)T \) |
| good | 7 | \( 1 + (-0.396 - 3.76i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.50 + 1.55i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.20 - 5.67i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.94 + 1.31i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-6.45 - 1.37i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-5.27 - 3.82i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.71 - 8.34i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.61 + 1.51i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.97 + 4.48i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 6.13i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 3.36i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.652 + 6.21i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.01 - 2.71i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 + (2.41 + 4.18i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.247 - 0.0260i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-2.26 + 5.08i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.661 - 1.48i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.71 + 1.90i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (0.635 - 0.461i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.23 - 0.900i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.650177064243383408814372561861, −9.037040386071013902879040709512, −8.641360660942297908645337642705, −7.61465824921468339641875967149, −6.94325144019183013627413026013, −5.76698189643182529065842942201, −5.28360680500292382165210425159, −3.29256105788903962768103773825, −2.35690647785828778443060505553, −1.32560273050445072731755716214,
1.00101398909273321791088347530, 2.74554309559518471314021225678, 3.36081616466756256782383082022, 4.66060626550796616562310614502, 5.54052010200472414001549650094, 7.02797003003426368401206384874, 7.80409129449519601320716327724, 8.229253093455594745352684680391, 9.473338185690471073391644960356, 10.22669598745229742043894996069
