| L(s) = 1 | + (0.5 − 0.866i)2-s + (1.13 + 1.31i)3-s + (−0.499 − 0.866i)4-s + (−1.60 + 1.55i)5-s + (1.70 − 0.322i)6-s + (−1.24 − 2.33i)7-s − 0.999·8-s + (−0.444 + 2.96i)9-s + (0.543 + 2.16i)10-s + 5.27i·11-s + (0.571 − 1.63i)12-s + (−2.80 + 4.86i)13-s + (−2.64 − 0.0856i)14-s + (−3.85 − 0.350i)15-s + (−0.5 + 0.866i)16-s + (4.30 + 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.652 + 0.757i)3-s + (−0.249 − 0.433i)4-s + (−0.718 + 0.695i)5-s + (0.694 − 0.131i)6-s + (−0.471 − 0.881i)7-s − 0.353·8-s + (−0.148 + 0.988i)9-s + (0.171 + 0.685i)10-s + 1.59i·11-s + (0.164 − 0.472i)12-s + (−0.778 + 1.34i)13-s + (−0.706 − 0.0228i)14-s + (−0.995 − 0.0904i)15-s + (−0.125 + 0.216i)16-s + (1.04 + 0.603i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10335 + 0.941653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10335 + 0.941653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.13 - 1.31i)T \) |
| 5 | \( 1 + (1.60 - 1.55i)T \) |
| 7 | \( 1 + (1.24 + 2.33i)T \) |
| good | 11 | \( 1 - 5.27iT - 11T^{2} \) |
| 13 | \( 1 + (2.80 - 4.86i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.30 - 2.48i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0258 - 0.0149i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 + (-6.48 + 3.74i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.73 + 3.88i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.82 - 1.05i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.423 - 0.733i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (7.31 - 4.22i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.11 - 1.79i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.914 + 1.58i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.21 - 2.11i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.66 - 2.11i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.3 - 7.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.23 + 3.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.11 + 3.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.84 + 5.68i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.09 - 8.83i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.01 + 10.4i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.45809210492264472588975436660, −10.06780114373917769987859777949, −9.528724930180940799080959874761, −8.110372029966484523471703735144, −7.35339901764731038293827356083, −6.40343865870407502648514397473, −4.53970430332683391951597868450, −4.29634510889722612275256007012, −3.22059301109848291403123606560, −2.09100299764311499786074543163,
0.65085053150951831240763107998, 2.89818267100171362257655390234, 3.45825167468637676441989064484, 5.13222856465192352736900536555, 5.83115736335215158454837907655, 6.88835061894068153653913861950, 8.043186809366129262262259010144, 8.285835014755311622197388946026, 9.110479117062983399934367773010, 10.22966563591780867090294322935
