| L(s) = 1 | + (14.5 − 44.9i)2-s + (9.99e2 − 726. i)3-s + (4.82e3 + 3.50e3i)4-s + (−1.80e4 − 5.54e4i)5-s + (−1.80e4 − 5.54e4i)6-s + (3.14e5 + 2.28e5i)7-s + (5.40e5 − 3.92e5i)8-s + (−2.07e4 + 6.37e4i)9-s − 2.75e6·10-s + (−3.98e6 − 4.31e6i)11-s + 7.36e6·12-s + (3.79e6 − 1.16e7i)13-s + (1.48e7 − 1.07e7i)14-s + (−5.83e7 − 4.23e7i)15-s + (5.34e6 + 1.64e7i)16-s + (−3.77e7 − 1.16e8i)17-s + ⋯ |
| L(s) = 1 | + (0.161 − 0.496i)2-s + (0.791 − 0.575i)3-s + (0.588 + 0.427i)4-s + (−0.515 − 1.58i)5-s + (−0.157 − 0.485i)6-s + (1.00 + 0.733i)7-s + (0.729 − 0.529i)8-s + (−0.0130 + 0.0400i)9-s − 0.870·10-s + (−0.678 − 0.734i)11-s + 0.712·12-s + (0.218 − 0.671i)13-s + (0.526 − 0.382i)14-s + (−1.32 − 0.960i)15-s + (0.0796 + 0.245i)16-s + (−0.378 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.83290 - 2.13305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83290 - 2.13305i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (3.98e6 + 4.31e6i)T \) |
| good | 2 | \( 1 + (-14.5 + 44.9i)T + (-6.62e3 - 4.81e3i)T^{2} \) |
| 3 | \( 1 + (-9.99e2 + 726. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (1.80e4 + 5.54e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-3.14e5 - 2.28e5i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-3.79e6 + 1.16e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (3.77e7 + 1.16e8i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (8.95e7 - 6.50e7i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 - 2.80e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-2.66e9 - 1.93e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-1.33e6 + 4.10e6i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-2.26e10 - 1.64e10i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (3.90e10 - 2.83e10i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 1.73e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-4.85e10 + 3.53e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (7.50e10 - 2.31e11i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (1.45e11 + 1.05e11i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-1.18e10 - 3.63e10i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 - 1.22e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (1.51e11 + 4.67e11i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (1.12e9 + 8.13e8i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (1.50e11 - 4.62e11i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-4.08e11 - 1.25e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 + 5.09e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-3.03e12 + 9.34e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−16.68738902368370109283005211988, −15.52754892351389342708291218275, −13.50215469901768817018434236520, −12.49440261557529903434530755137, −11.29260439169864453601460375145, −8.547686019847246451321807520247, −7.913718344027953237841331005167, −4.94253854302669381064258576125, −2.73112910156441734835538827143, −1.21412120034000522292646157083,
2.31569667850402573600011028229, 4.17443466047977011839045333228, 6.68569139721936476204422463054, 7.911705141345256440328375381856, 10.34044764860762217987147934386, 11.20487621404505138843360110235, 14.13700316210284254768389870786, 14.85830241752013711875712412776, 15.55503443397426202379945884519, 17.56101883691164929748457786325
