| L(s) = 1 | + (4.40 − 13.5i)2-s + (169. − 122. i)3-s + (249. + 181. i)4-s + (63.1 + 194. i)5-s + (−920. − 2.83e3i)6-s + (−1.58e3 − 1.15e3i)7-s + (9.46e3 − 6.87e3i)8-s + (7.42e3 − 2.28e4i)9-s + 2.91e3·10-s + (−4.62e4 − 1.49e4i)11-s + 6.46e4·12-s + (−1.67e4 + 5.16e4i)13-s + (−2.25e4 + 1.64e4i)14-s + (3.45e4 + 2.51e4i)15-s + (−2.63e3 − 8.09e3i)16-s + (1.24e5 + 3.83e5i)17-s + ⋯ |
| L(s) = 1 | + (0.194 − 0.598i)2-s + (1.20 − 0.876i)3-s + (0.488 + 0.354i)4-s + (0.0451 + 0.139i)5-s + (−0.290 − 0.892i)6-s + (−0.249 − 0.181i)7-s + (0.816 − 0.593i)8-s + (0.377 − 1.16i)9-s + 0.0921·10-s + (−0.951 − 0.306i)11-s + 0.899·12-s + (−0.162 + 0.501i)13-s + (−0.157 + 0.114i)14-s + (0.176 + 0.128i)15-s + (−0.0100 − 0.0308i)16-s + (0.362 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.354 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.17417 - 1.50128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17417 - 1.50128i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.62e4 + 1.49e4i)T \) |
| good | 2 | \( 1 + (-4.40 + 13.5i)T + (-414. - 300. i)T^{2} \) |
| 3 | \( 1 + (-169. + 122. i)T + (6.08e3 - 1.87e4i)T^{2} \) |
| 5 | \( 1 + (-63.1 - 194. i)T + (-1.58e6 + 1.14e6i)T^{2} \) |
| 7 | \( 1 + (1.58e3 + 1.15e3i)T + (1.24e7 + 3.83e7i)T^{2} \) |
| 13 | \( 1 + (1.67e4 - 5.16e4i)T + (-8.57e9 - 6.23e9i)T^{2} \) |
| 17 | \( 1 + (-1.24e5 - 3.83e5i)T + (-9.59e10 + 6.97e10i)T^{2} \) |
| 19 | \( 1 + (4.33e5 - 3.14e5i)T + (9.97e10 - 3.06e11i)T^{2} \) |
| 23 | \( 1 - 1.65e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (4.60e6 + 3.34e6i)T + (4.48e12 + 1.37e13i)T^{2} \) |
| 31 | \( 1 + (2.77e6 - 8.53e6i)T + (-2.13e13 - 1.55e13i)T^{2} \) |
| 37 | \( 1 + (3.75e6 + 2.73e6i)T + (4.01e13 + 1.23e14i)T^{2} \) |
| 41 | \( 1 + (6.01e6 - 4.36e6i)T + (1.01e14 - 3.11e14i)T^{2} \) |
| 43 | \( 1 + 2.24e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-3.63e7 + 2.64e7i)T + (3.45e14 - 1.06e15i)T^{2} \) |
| 53 | \( 1 + (-5.04e6 + 1.55e7i)T + (-2.66e15 - 1.93e15i)T^{2} \) |
| 59 | \( 1 + (-3.55e7 - 2.58e7i)T + (2.67e15 + 8.23e15i)T^{2} \) |
| 61 | \( 1 + (4.25e7 + 1.30e8i)T + (-9.46e15 + 6.87e15i)T^{2} \) |
| 67 | \( 1 + 2.31e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-5.56e7 - 1.71e8i)T + (-3.70e16 + 2.69e16i)T^{2} \) |
| 73 | \( 1 + (-5.65e7 - 4.10e7i)T + (1.81e16 + 5.59e16i)T^{2} \) |
| 79 | \( 1 + (-1.49e8 + 4.61e8i)T + (-9.69e16 - 7.04e16i)T^{2} \) |
| 83 | \( 1 + (-5.27e7 - 1.62e8i)T + (-1.51e17 + 1.09e17i)T^{2} \) |
| 89 | \( 1 - 3.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-3.92e7 + 1.20e8i)T + (-6.15e17 - 4.46e17i)T^{2} \) |
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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
−18.71451443935224171529378615045, −16.67503746964658028551550703882, −14.88882352893269136049501337061, −13.37692742088982714101608196050, −12.52474885153764305262175472352, −10.59082165803249876809143091298, −8.379525753719441216662588691171, −7.00856998814705610400837977469, −3.32177356130365272727492673682, −1.89658651694402792827196504236,
2.72565283510831581272714660628, 5.09155508649033647533579290150, 7.48643625609591347041551203470, 9.230309149002564037382312720865, 10.71871024758409645409399377882, 13.27393678362511741326794810964, 14.82934394025471181630174047448, 15.40613415060751819131676729150, 16.65302118639729355890989171437, 18.88663848090359386352849156320
