| L(s) = 1 | + (35.8 − 26.0i)2-s + (63.0 + 194. i)3-s + (447. − 1.37e3i)4-s + (376. + 273. i)5-s + (7.30e3 + 5.30e3i)6-s + (−442. + 1.36e3i)7-s + (−1.27e4 − 3.93e4i)8-s + (−1.77e4 + 1.29e4i)9-s + 2.06e4·10-s + (−4.84e4 − 2.64e3i)11-s + 2.95e5·12-s + (−6.45e4 + 4.68e4i)13-s + (1.96e4 + 6.03e4i)14-s + (−2.93e4 + 9.03e4i)15-s + (−8.83e5 − 6.42e5i)16-s + (2.91e5 + 2.12e5i)17-s + ⋯ |
| L(s) = 1 | + (1.58 − 1.14i)2-s + (0.449 + 1.38i)3-s + (0.873 − 2.68i)4-s + (0.269 + 0.195i)5-s + (2.30 + 1.67i)6-s + (−0.0697 + 0.214i)7-s + (−1.10 − 3.40i)8-s + (−0.902 + 0.655i)9-s + 0.651·10-s + (−0.998 − 0.0543i)11-s + 4.11·12-s + (−0.626 + 0.455i)13-s + (0.136 + 0.419i)14-s + (−0.149 + 0.460i)15-s + (−3.37 − 2.44i)16-s + (0.847 + 0.615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.56538 - 1.31460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.56538 - 1.31460i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.84e4 + 2.64e3i)T \) |
| good | 2 | \( 1 + (-35.8 + 26.0i)T + (158. - 486. i)T^{2} \) |
| 3 | \( 1 + (-63.0 - 194. i)T + (-1.59e4 + 1.15e4i)T^{2} \) |
| 5 | \( 1 + (-376. - 273. i)T + (6.03e5 + 1.85e6i)T^{2} \) |
| 7 | \( 1 + (442. - 1.36e3i)T + (-3.26e7 - 2.37e7i)T^{2} \) |
| 13 | \( 1 + (6.45e4 - 4.68e4i)T + (3.27e9 - 1.00e10i)T^{2} \) |
| 17 | \( 1 + (-2.91e5 - 2.12e5i)T + (3.66e10 + 1.12e11i)T^{2} \) |
| 19 | \( 1 + (-8.51e4 - 2.62e5i)T + (-2.61e11 + 1.89e11i)T^{2} \) |
| 23 | \( 1 + 1.57e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + (-2.75e5 + 8.49e5i)T + (-1.17e13 - 8.52e12i)T^{2} \) |
| 31 | \( 1 + (-1.84e6 + 1.34e6i)T + (8.17e12 - 2.51e13i)T^{2} \) |
| 37 | \( 1 + (-2.38e6 + 7.32e6i)T + (-1.05e14 - 7.63e13i)T^{2} \) |
| 41 | \( 1 + (5.47e6 + 1.68e7i)T + (-2.64e14 + 1.92e14i)T^{2} \) |
| 43 | \( 1 - 5.17e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.18e7 - 3.63e7i)T + (-9.05e14 + 6.57e14i)T^{2} \) |
| 53 | \( 1 + (-6.96e7 + 5.05e7i)T + (1.01e15 - 3.13e15i)T^{2} \) |
| 59 | \( 1 + (2.81e7 - 8.67e7i)T + (-7.00e15 - 5.09e15i)T^{2} \) |
| 61 | \( 1 + (6.38e7 + 4.63e7i)T + (3.61e15 + 1.11e16i)T^{2} \) |
| 67 | \( 1 - 1.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-1.37e8 - 9.99e7i)T + (1.41e16 + 4.36e16i)T^{2} \) |
| 73 | \( 1 + (4.61e7 - 1.42e8i)T + (-4.76e16 - 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-1.82e8 + 1.32e8i)T + (3.70e16 - 1.13e17i)T^{2} \) |
| 83 | \( 1 + (5.17e8 + 3.75e8i)T + (5.77e16 + 1.77e17i)T^{2} \) |
| 89 | \( 1 + 1.02e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (7.94e8 - 5.76e8i)T + (2.34e17 - 7.23e17i)T^{2} \) |
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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
−18.91460577084794950408190151884, −15.97623401759944566682163497882, −14.91069477962793113231570482119, −13.95210037617886339474165014808, −12.31133723914866419568323151149, −10.62418317680722463632546306877, −9.788985630090164224384632687475, −5.54981572517781857368700957341, −4.09682528298098833474831177845, −2.59645559016435511657230668747,
2.77456850123899318428511317921, 5.33328438103264427328311786327, 7.03991211789358857456932425776, 7.992225134165057690419170850105, 12.11572577114543346717409629968, 13.14772079551227386609270992603, 13.87851391896760685537796189327, 15.27288300312912402692549889861, 16.84233557280918110451425174896, 18.16525170672542196901334119791
