| L(s) = 1 | + (0.837 − 2.13i)2-s + (−0.00927 + 1.73i)3-s + (−2.38 − 2.21i)4-s + (−0.0993 + 0.0478i)5-s + (3.68 + 1.47i)6-s + (2.41 − 1.08i)7-s + (−2.60 + 1.25i)8-s + (−2.99 − 0.0321i)9-s + (0.0189 + 0.252i)10-s + (1.61 + 2.01i)11-s + (3.86 − 4.11i)12-s + (1.40 − 3.57i)13-s + (−0.294 − 6.06i)14-s + (−0.0819 − 0.172i)15-s + (0.00775 + 0.103i)16-s + (5.78 − 5.36i)17-s + ⋯ |
| L(s) = 1 | + (0.592 − 1.50i)2-s + (−0.00535 + 0.999i)3-s + (−1.19 − 1.10i)4-s + (−0.0444 + 0.0214i)5-s + (1.50 + 0.600i)6-s + (0.912 − 0.410i)7-s + (−0.919 + 0.442i)8-s + (−0.999 − 0.0107i)9-s + (0.00597 + 0.0797i)10-s + (0.485 + 0.608i)11-s + (1.11 − 1.18i)12-s + (0.389 − 0.992i)13-s + (−0.0786 − 1.61i)14-s + (−0.0211 − 0.0445i)15-s + (0.00193 + 0.0258i)16-s + (1.40 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0400 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0400 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42620 - 1.37019i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42620 - 1.37019i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.00927 - 1.73i)T \) |
| 7 | \( 1 + (-2.41 + 1.08i)T \) |
| good | 2 | \( 1 + (-0.837 + 2.13i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (0.0993 - 0.0478i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-1.61 - 2.01i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.40 + 3.57i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-5.78 + 5.36i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.178 + 0.309i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.551 - 2.41i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-9.38 - 2.89i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (2.98 - 5.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (8.76 + 2.70i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (0.0868 + 0.0592i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (10.3 - 7.02i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-1.47 + 3.75i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-1.40 + 0.434i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (1.36 - 0.931i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (5.31 - 4.93i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (2.39 - 4.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.46 - 6.43i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.438 - 0.0661i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-3.04 - 5.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.45 - 3.71i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-4.67 - 11.9i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-6.18 + 10.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.90047848539546531595153766057, −10.23811962505598519805297678786, −9.637494242873150767008498722291, −8.513948836801639074715690348661, −7.31240381781208032174037415945, −5.35391064457256555522402909109, −4.92150727134931141035163696735, −3.74435447487862022165095421733, −3.00738349426666339654236063399, −1.30850496305617520776626643495,
1.72359953227906825264625452905, 3.70960892981310298776147240924, 4.95326715587439528402429513484, 6.06301476542776260076743846118, 6.40613169650802435626848389080, 7.64376116581474046554278204128, 8.278295284670791753713914533316, 8.810108983503866882204212647695, 10.54989913022800791213156953591, 11.85024591112092353743622017596
