| L(s) = 1 | + (−27.1 − 8.81i)2-s + (102. − 74.1i)3-s + (451. + 328. i)4-s + (−273. − 840. i)5-s + (−3.42e3 + 1.11e3i)6-s + (−623. + 857. i)7-s + (−5.07e3 − 6.98e3i)8-s + (2.88e3 − 8.89e3i)9-s + 2.52e4i·10-s + (−1.46e4 + 204. i)11-s + 7.04e4·12-s + (−8.77e3 − 2.85e3i)13-s + (2.44e4 − 1.77e4i)14-s + (−9.01e4 − 6.55e4i)15-s + (3.19e4 + 9.82e4i)16-s + (5.54e4 − 1.80e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.69 − 0.551i)2-s + (1.25 − 0.915i)3-s + (1.76 + 1.28i)4-s + (−0.436 − 1.34i)5-s + (−2.64 + 0.858i)6-s + (−0.259 + 0.357i)7-s + (−1.23 − 1.70i)8-s + (0.440 − 1.35i)9-s + 2.52i·10-s + (−0.999 + 0.0139i)11-s + 3.39·12-s + (−0.307 − 0.0998i)13-s + (0.637 − 0.463i)14-s + (−1.78 − 1.29i)15-s + (0.487 + 1.49i)16-s + (0.663 − 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.117293 - 0.760285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.117293 - 0.760285i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (1.46e4 - 204. i)T \) |
| good | 2 | \( 1 + (27.1 + 8.81i)T + (207. + 150. i)T^{2} \) |
| 3 | \( 1 + (-102. + 74.1i)T + (2.02e3 - 6.23e3i)T^{2} \) |
| 5 | \( 1 + (273. + 840. i)T + (-3.16e5 + 2.29e5i)T^{2} \) |
| 7 | \( 1 + (623. - 857. i)T + (-1.78e6 - 5.48e6i)T^{2} \) |
| 13 | \( 1 + (8.77e3 + 2.85e3i)T + (6.59e8 + 4.79e8i)T^{2} \) |
| 17 | \( 1 + (-5.54e4 + 1.80e4i)T + (5.64e9 - 4.10e9i)T^{2} \) |
| 19 | \( 1 + (6.71e3 + 9.24e3i)T + (-5.24e9 + 1.61e10i)T^{2} \) |
| 23 | \( 1 - 6.17e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-4.30e5 + 5.92e5i)T + (-1.54e11 - 4.75e11i)T^{2} \) |
| 31 | \( 1 + (-5.48e5 + 1.68e6i)T + (-6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.91e6 - 1.39e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (1.81e6 + 2.49e6i)T + (-2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 4.50e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.28e6 + 1.66e6i)T + (7.35e12 - 2.26e13i)T^{2} \) |
| 53 | \( 1 + (-4.01e5 + 1.23e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (1.39e5 + 1.01e5i)T + (4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.67e7 + 5.42e6i)T + (1.55e14 - 1.12e14i)T^{2} \) |
| 67 | \( 1 + 2.03e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-9.68e6 - 2.98e7i)T + (-5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (-2.51e7 + 3.46e7i)T + (-2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (1.49e7 + 4.87e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (7.13e7 - 2.31e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 - 2.98e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (1.38e7 - 4.25e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−18.53314856812837154051998458663, −16.98059207542072425281627502819, −15.59191334279728406134265793981, −13.13759049607849726207866032457, −12.01857280399458176736182624013, −9.651602951380973991890270983523, −8.439558082408893720327464139450, −7.71124130947749563882157030001, −2.50035536652065916076456778779, −0.74490861116489186716859111086,
2.93121310570515078976480940422, 7.14648105411443798039875919162, 8.354598826112155576640872955439, 9.955227239805415185003159935307, 10.67785850366264761779875818564, 14.37923928408927055688534597669, 15.32066341751430370609803099919, 16.28107493539408092202723271306, 18.12190562282371857835357343188, 19.19125666696763247344298691721
