| L(s) = 1 | + (0.741 + 0.670i)2-s + (−0.119 + 2.58i)3-s + (0.100 + 0.994i)4-s + (−1.33 − 1.50i)5-s + (−1.82 + 1.84i)6-s + (−3.40 − 0.962i)7-s + (−0.592 + 0.805i)8-s + (−3.70 − 0.341i)9-s + (0.0185 − 2.01i)10-s + (1.36 − 0.641i)11-s + (−2.58 + 0.142i)12-s + (−3.83 + 0.495i)13-s + (−1.87 − 2.99i)14-s + (4.06 − 3.28i)15-s + (−0.979 + 0.200i)16-s + (−2.11 + 0.952i)17-s + ⋯ |
| L(s) = 1 | + (0.524 + 0.474i)2-s + (−0.0687 + 1.49i)3-s + (0.0504 + 0.497i)4-s + (−0.598 − 0.674i)5-s + (−0.744 + 0.751i)6-s + (−1.28 − 0.363i)7-s + (−0.209 + 0.284i)8-s + (−1.23 − 0.113i)9-s + (0.00585 − 0.637i)10-s + (0.410 − 0.193i)11-s + (−0.747 + 0.0412i)12-s + (−1.06 + 0.137i)13-s + (−0.502 − 0.800i)14-s + (1.04 − 0.848i)15-s + (−0.244 + 0.0501i)16-s + (−0.513 + 0.231i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132854 - 0.174291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132854 - 0.174291i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.741 - 0.670i)T \) |
| 19 | \( 1 + (-2.47 + 3.58i)T \) |
| good | 3 | \( 1 + (0.119 - 2.58i)T + (-2.98 - 0.275i)T^{2} \) |
| 5 | \( 1 + (1.33 + 1.50i)T + (-0.595 + 4.96i)T^{2} \) |
| 7 | \( 1 + (3.40 + 0.962i)T + (5.96 + 3.66i)T^{2} \) |
| 11 | \( 1 + (-1.36 + 0.641i)T + (6.99 - 8.49i)T^{2} \) |
| 13 | \( 1 + (3.83 - 0.495i)T + (12.5 - 3.30i)T^{2} \) |
| 17 | \( 1 + (2.11 - 0.952i)T + (11.2 - 12.7i)T^{2} \) |
| 23 | \( 1 + (0.388 + 0.200i)T + (13.2 + 18.7i)T^{2} \) |
| 29 | \( 1 + (8.21 - 0.604i)T + (28.6 - 4.24i)T^{2} \) |
| 31 | \( 1 + (1.92 - 0.781i)T + (22.2 - 21.6i)T^{2} \) |
| 37 | \( 1 + (0.797 - 9.62i)T + (-36.4 - 6.08i)T^{2} \) |
| 41 | \( 1 + (-0.282 + 0.468i)T + (-19.1 - 36.2i)T^{2} \) |
| 43 | \( 1 + (-0.849 - 0.479i)T + (22.1 + 36.8i)T^{2} \) |
| 47 | \( 1 + (1.21 + 0.319i)T + (40.9 + 23.1i)T^{2} \) |
| 53 | \( 1 + (-1.78 - 6.54i)T + (-45.6 + 26.9i)T^{2} \) |
| 59 | \( 1 + (0.307 + 0.555i)T + (-31.3 + 49.9i)T^{2} \) |
| 61 | \( 1 + (-0.863 + 13.4i)T + (-60.4 - 7.82i)T^{2} \) |
| 67 | \( 1 + (15.1 - 3.69i)T + (59.5 - 30.8i)T^{2} \) |
| 71 | \( 1 + (-0.410 - 6.37i)T + (-70.4 + 9.10i)T^{2} \) |
| 73 | \( 1 + (-10.0 - 7.22i)T + (23.0 + 69.2i)T^{2} \) |
| 79 | \( 1 + (6.99 - 4.12i)T + (38.2 - 69.1i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 6.91i)T + (-82.8 - 4.57i)T^{2} \) |
| 89 | \( 1 + (7.01 - 5.05i)T + (28.1 - 84.4i)T^{2} \) |
| 97 | \( 1 + (-6.85 - 1.66i)T + (86.1 + 44.5i)T^{2} \) |
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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
−10.99744513255154497394226684130, −9.932731045359667728284840008395, −9.387149578326220875620216540592, −8.655124750423597715568621288798, −7.42121968499329303993296339028, −6.53119501274039645836712964338, −5.36691146518483292802489705683, −4.56333846420899039660788949181, −3.88269381850629750164480966546, −3.00637951523303851803413920551,
0.089793123114264075684597330065, 1.94093331751841366470586044436, 2.92974130194680422703442772824, 3.87464835173278314866983968275, 5.53525170749389578883583556299, 6.32000810536961642607322992721, 7.22499045505492065589081677981, 7.53858583130206325479789974131, 9.076170126250326772906570240030, 9.833426788782665444848659755598
