| L(s) = 1 | + (22.7 + 70.1i)2-s + (−1.61e3 − 1.17e3i)3-s + (2.22e3 − 1.61e3i)4-s + (1.85e3 − 5.71e3i)5-s + (4.56e4 − 1.40e5i)6-s + (−1.79e5 + 1.30e5i)7-s + (6.53e5 + 4.74e5i)8-s + (7.45e5 + 2.29e6i)9-s + 4.43e5·10-s + (−2.68e6 + 5.22e6i)11-s − 5.50e6·12-s + (−2.42e6 − 7.47e6i)13-s + (−1.32e7 − 9.60e6i)14-s + (−9.73e6 + 7.07e6i)15-s + (−1.14e7 + 3.52e7i)16-s + (−4.64e7 + 1.42e8i)17-s + ⋯ |
| L(s) = 1 | + (0.251 + 0.775i)2-s + (−1.28 − 0.931i)3-s + (0.271 − 0.197i)4-s + (0.0531 − 0.163i)5-s + (0.399 − 1.22i)6-s + (−0.575 + 0.418i)7-s + (0.880 + 0.639i)8-s + (0.467 + 1.43i)9-s + 0.140·10-s + (−0.456 + 0.889i)11-s − 0.531·12-s + (−0.139 − 0.429i)13-s + (−0.469 − 0.340i)14-s + (−0.220 + 0.160i)15-s + (−0.170 + 0.524i)16-s + (−0.466 + 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.497859 + 0.769462i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497859 + 0.769462i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (2.68e6 - 5.22e6i)T \) |
| good | 2 | \( 1 + (-22.7 - 70.1i)T + (-6.62e3 + 4.81e3i)T^{2} \) |
| 3 | \( 1 + (1.61e3 + 1.17e3i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-1.85e3 + 5.71e3i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (1.79e5 - 1.30e5i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (2.42e6 + 7.47e6i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (4.64e7 - 1.42e8i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-2.50e8 - 1.81e8i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 + 4.49e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (2.89e8 - 2.09e8i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (1.04e9 + 3.22e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (4.32e9 - 3.14e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (1.82e10 + 1.32e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 + 1.80e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (9.20e10 + 6.68e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-8.06e10 - 2.48e11i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (6.17e10 - 4.48e10i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-4.16e10 + 1.28e11i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 + 1.12e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (5.80e11 - 1.78e12i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-1.75e12 + 1.27e12i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (7.54e11 + 2.32e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (3.59e11 - 1.10e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 - 8.58e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (2.43e11 + 7.49e11i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−17.45106138986207541459844292352, −16.38805015221376168317443256827, −15.10699209704411927181347724022, −13.14242751716003596975150064207, −12.03234763105176421866940829568, −10.42739575188230103422458181372, −7.60471319999438027529718038642, −6.31923280830342866642042623179, −5.31745871136873790616434400503, −1.64299967486993899492900712540,
0.43416584773340433620746699414, 3.18079507923692913401623362864, 4.87682907942107424469607892880, 6.79003967628619181750977563415, 9.824022128374192902944053072561, 11.00008838197877122961325949605, 11.82921988366394952160450458179, 13.53609107818178500344018255771, 16.06180358324203815461565508348, 16.35871638065147525109944884473
