| L(s) = 1 | + (0.926 + 0.376i)2-s + (2.59 + 0.580i)3-s + (0.716 + 0.697i)4-s + (0.0939 + 3.40i)5-s + (2.18 + 1.51i)6-s + (−0.993 − 3.92i)7-s + (0.401 + 0.915i)8-s + (3.66 + 1.73i)9-s + (−1.19 + 3.19i)10-s + (−0.317 + 3.83i)11-s + (1.45 + 2.22i)12-s + (−0.511 − 1.63i)13-s + (0.556 − 4.00i)14-s + (−1.73 + 8.89i)15-s + (0.0275 + 0.999i)16-s + (−0.772 + 0.751i)17-s + ⋯ |
| L(s) = 1 | + (0.655 + 0.266i)2-s + (1.49 + 0.335i)3-s + (0.358 + 0.348i)4-s + (0.0420 + 1.52i)5-s + (0.891 + 0.617i)6-s + (−0.375 − 1.48i)7-s + (0.142 + 0.323i)8-s + (1.22 + 0.577i)9-s + (−0.378 + 1.01i)10-s + (−0.0957 + 1.15i)11-s + (0.419 + 0.641i)12-s + (−0.141 − 0.453i)13-s + (0.148 − 1.07i)14-s + (−0.448 + 2.29i)15-s + (0.00688 + 0.249i)16-s + (−0.187 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.79470 + 1.96345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79470 + 1.96345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.926 - 0.376i)T \) |
| 19 | \( 1 + (1.71 + 4.00i)T \) |
| good | 3 | \( 1 + (-2.59 - 0.580i)T + (2.71 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.0939 - 3.40i)T + (-4.99 + 0.275i)T^{2} \) |
| 7 | \( 1 + (0.993 + 3.92i)T + (-6.15 + 3.33i)T^{2} \) |
| 11 | \( 1 + (0.317 - 3.83i)T + (-10.8 - 1.81i)T^{2} \) |
| 13 | \( 1 + (0.511 + 1.63i)T + (-10.6 + 7.40i)T^{2} \) |
| 17 | \( 1 + (0.772 - 0.751i)T + (0.468 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-4.20 + 0.941i)T + (20.8 - 9.81i)T^{2} \) |
| 29 | \( 1 + (-8.47 - 2.39i)T + (24.7 + 15.1i)T^{2} \) |
| 31 | \( 1 + (2.97 + 2.31i)T + (7.61 + 30.0i)T^{2} \) |
| 37 | \( 1 + (-0.233 + 2.82i)T + (-36.4 - 6.08i)T^{2} \) |
| 41 | \( 1 + (4.23 + 3.68i)T + (5.63 + 40.6i)T^{2} \) |
| 43 | \( 1 + (0.539 + 1.44i)T + (-32.4 + 28.2i)T^{2} \) |
| 47 | \( 1 + (7.86 + 5.45i)T + (16.4 + 44.0i)T^{2} \) |
| 53 | \( 1 + (-6.57 - 4.55i)T + (18.5 + 49.6i)T^{2} \) |
| 59 | \( 1 + (5.59 + 4.87i)T + (8.10 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-7.33 - 9.97i)T + (-18.2 + 58.2i)T^{2} \) |
| 67 | \( 1 + (6.11 + 0.676i)T + (65.3 + 14.6i)T^{2} \) |
| 71 | \( 1 + (-3.57 + 4.85i)T + (-21.1 - 67.7i)T^{2} \) |
| 73 | \( 1 + (-6.45 + 6.28i)T + (2.01 - 72.9i)T^{2} \) |
| 79 | \( 1 + (0.745 + 1.99i)T + (-59.5 + 51.8i)T^{2} \) |
| 83 | \( 1 + (8.84 - 4.78i)T + (45.3 - 69.4i)T^{2} \) |
| 89 | \( 1 + (-8.96 - 8.72i)T + (2.45 + 88.9i)T^{2} \) |
| 97 | \( 1 + (7.63 - 0.845i)T + (94.6 - 21.2i)T^{2} \) |
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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
−10.37156781616131465877175331799, −9.975560519381299669537935939859, −8.789243185375965993312874669990, −7.64451982927866545761461284945, −7.10481260133641321533422055399, −6.60804872396605065517011089509, −4.76893292779903218770528356949, −3.86060284662910663867824992267, −3.12184150480218551369086332250, −2.30717349710369532877140999307,
1.46674312799206911946934112414, 2.59304590925068273959942398144, 3.43526176738425992889352419532, 4.70218403876159680178481974828, 5.59564184271435643205166528284, 6.56663917899004106258278279889, 8.125099048682502013668784744841, 8.554839795712718580624107650108, 9.122908841386527434945276014269, 9.891021839751160963324310270101
