L(s) = 1 | + (73.1 + 23.7i)2-s + (491. − 357. i)3-s + (1.46e3 + 1.06e3i)4-s + (−9.39e3 − 2.89e4i)5-s + (4.44e4 − 1.44e4i)6-s + (2.66e4 − 3.67e4i)7-s + (−1.03e5 − 1.41e5i)8-s + (−5.00e4 + 1.53e5i)9-s − 2.33e6i·10-s + (2.16e5 + 1.75e6i)11-s + 1.10e6·12-s + (6.06e6 + 1.96e6i)13-s + (2.82e6 − 2.04e6i)14-s + (−1.49e7 − 1.08e7i)15-s + (−6.46e6 − 1.98e7i)16-s + (3.22e7 − 1.04e7i)17-s + ⋯ |
L(s) = 1 | + (1.14 + 0.371i)2-s + (0.674 − 0.490i)3-s + (0.358 + 0.260i)4-s + (−0.601 − 1.85i)5-s + (0.952 − 0.309i)6-s + (0.226 − 0.311i)7-s + (−0.393 − 0.541i)8-s + (−0.0941 + 0.289i)9-s − 2.33i·10-s + (0.122 + 0.992i)11-s + 0.369·12-s + (1.25 + 0.408i)13-s + (0.374 − 0.272i)14-s + (−1.31 − 0.953i)15-s + (−0.385 − 1.18i)16-s + (1.33 − 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.75739 - 1.79880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.75739 - 1.79880i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-2.16e5 - 1.75e6i)T \) |
good | 2 | \( 1 + (-73.1 - 23.7i)T + (3.31e3 + 2.40e3i)T^{2} \) |
| 3 | \( 1 + (-491. + 357. i)T + (1.64e5 - 5.05e5i)T^{2} \) |
| 5 | \( 1 + (9.39e3 + 2.89e4i)T + (-1.97e8 + 1.43e8i)T^{2} \) |
| 7 | \( 1 + (-2.66e4 + 3.67e4i)T + (-4.27e9 - 1.31e10i)T^{2} \) |
| 13 | \( 1 + (-6.06e6 - 1.96e6i)T + (1.88e13 + 1.36e13i)T^{2} \) |
| 17 | \( 1 + (-3.22e7 + 1.04e7i)T + (4.71e14 - 3.42e14i)T^{2} \) |
| 19 | \( 1 + (-3.76e6 - 5.17e6i)T + (-6.83e14 + 2.10e15i)T^{2} \) |
| 23 | \( 1 - 7.05e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (1.53e7 - 2.10e7i)T + (-1.09e17 - 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-2.23e8 + 6.89e8i)T + (-6.37e17 - 4.62e17i)T^{2} \) |
| 37 | \( 1 + (2.12e9 + 1.54e9i)T + (2.03e18 + 6.26e18i)T^{2} \) |
| 41 | \( 1 + (2.68e7 + 3.69e7i)T + (-6.97e18 + 2.14e19i)T^{2} \) |
| 43 | \( 1 + 2.66e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-6.45e9 + 4.69e9i)T + (3.59e19 - 1.10e20i)T^{2} \) |
| 53 | \( 1 + (5.66e9 - 1.74e10i)T + (-3.97e20 - 2.88e20i)T^{2} \) |
| 59 | \( 1 + (1.06e10 + 7.72e9i)T + (5.49e20 + 1.69e21i)T^{2} \) |
| 61 | \( 1 + (1.47e10 - 4.79e9i)T + (2.14e21 - 1.56e21i)T^{2} \) |
| 67 | \( 1 + 1.63e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-2.27e10 - 7.01e10i)T + (-1.32e22 + 9.64e21i)T^{2} \) |
| 73 | \( 1 + (6.31e10 - 8.69e10i)T + (-7.07e21 - 2.17e22i)T^{2} \) |
| 79 | \( 1 + (-2.70e11 - 8.77e10i)T + (4.78e22 + 3.47e22i)T^{2} \) |
| 83 | \( 1 + (2.47e11 - 8.03e10i)T + (8.64e22 - 6.28e22i)T^{2} \) |
| 89 | \( 1 - 3.97e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (3.24e11 - 1.00e12i)T + (-5.61e23 - 4.07e23i)T^{2} \) |
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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
−16.77848381499584804377287442076, −15.63672407076965584386909634427, −14.05804378543848448429256030597, −13.08347881201670356799890965508, −12.09450216297466000245905964595, −9.079899874159016277104209968866, −7.64713831853706500599124760040, −5.25584852009926543962581973904, −3.98623971637328269973267371414, −1.18094834150320351798487029060,
3.11273886820233483392673130413, 3.54556073161034126355080839949, 6.10144353613497103488768320588, 8.359907634625851303532571080116, 10.67122134207151087897153948155, 11.80674791930410520944969967582, 13.85608846854559956064968508553, 14.64981594267107739971565393127, 15.53063276747302357572290945786, 18.12263209442221036569586178399