| L(s) = 1 | + (1.06 + 0.957i)2-s + (−0.994 + 0.104i)3-s + (0.00498 + 0.0474i)4-s + (−1.89 + 1.17i)5-s + (−1.15 − 0.841i)6-s + (1.17 − 2.37i)7-s + (1.64 − 2.26i)8-s + (0.978 − 0.207i)9-s + (−3.14 − 0.564i)10-s + (2.22 + 0.472i)11-s + (−0.00992 − 0.0466i)12-s + (3.31 + 1.07i)13-s + (3.51 − 1.39i)14-s + (1.76 − 1.37i)15-s + (4.00 − 0.851i)16-s + (−0.282 − 0.634i)17-s + ⋯ |
| L(s) = 1 | + (0.751 + 0.677i)2-s + (−0.574 + 0.0603i)3-s + (0.00249 + 0.0237i)4-s + (−0.849 + 0.527i)5-s + (−0.472 − 0.343i)6-s + (0.444 − 0.895i)7-s + (0.580 − 0.799i)8-s + (0.326 − 0.0693i)9-s + (−0.996 − 0.178i)10-s + (0.669 + 0.142i)11-s + (−0.00286 − 0.0134i)12-s + (0.919 + 0.298i)13-s + (0.940 − 0.372i)14-s + (0.455 − 0.354i)15-s + (1.00 − 0.212i)16-s + (−0.0685 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73986 + 0.202443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73986 + 0.202443i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.994 - 0.104i)T \) |
| 5 | \( 1 + (1.89 - 1.17i)T \) |
| 7 | \( 1 + (-1.17 + 2.37i)T \) |
| good | 2 | \( 1 + (-1.06 - 0.957i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (-2.22 - 0.472i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.31 - 1.07i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.282 + 0.634i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.246 + 2.34i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-5.40 - 4.86i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 1.08i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.13 - 0.503i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.07 + 9.74i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (0.639 - 1.96i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.14iT - 43T^{2} \) |
| 47 | \( 1 + (1.15 - 2.58i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-5.07 + 0.533i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (4.07 + 4.52i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-7.14 + 7.92i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.82 - 4.09i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (6.14 - 4.46i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.403 - 1.90i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (14.8 + 6.62i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.09 + 4.25i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (10.7 - 11.9i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 15.3i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.12950494182422213215623953995, −10.26220199121241259571097952234, −9.115183329411285899505690888696, −7.73566378293153113479449063603, −7.01615028012821223664557806749, −6.42610536501036135616161782336, −5.21654881776668712738636076544, −4.28441423575982764155744516323, −3.60176112310823220237366987536, −1.10627362116079482746807498987,
1.43431100505413287297789035767, 3.07824352033211683892245538465, 4.12984365009303601759786500299, 4.96180572306268164971509169426, 5.86870279570494798064473204125, 7.16646611522094489390483262029, 8.476592655655976343991136880024, 8.675555104630036589803101096892, 10.36986824744344262423803213686, 11.28700108404088668474535079113
