| L(s) = 1 | + (2.53 + 0.824i)2-s + (83.6 − 60.7i)3-s + (−201. − 146. i)4-s + (23.5 + 72.4i)5-s + (262. − 85.2i)6-s + (2.43e3 − 3.34e3i)7-s + (−792. − 1.09e3i)8-s + (1.27e3 − 3.93e3i)9-s + 203. i·10-s + (1.54e3 + 1.45e4i)11-s − 2.57e4·12-s + (1.42e4 + 4.64e3i)13-s + (8.93e3 − 6.49e3i)14-s + (6.37e3 + 4.62e3i)15-s + (1.85e4 + 5.71e4i)16-s + (−1.10e5 + 3.57e4i)17-s + ⋯ |
| L(s) = 1 | + (0.158 + 0.0515i)2-s + (1.03 − 0.750i)3-s + (−0.786 − 0.571i)4-s + (0.0376 + 0.115i)5-s + (0.202 − 0.0658i)6-s + (1.01 − 1.39i)7-s + (−0.193 − 0.266i)8-s + (0.194 − 0.599i)9-s + 0.0203i·10-s + (0.105 + 0.994i)11-s − 1.24·12-s + (0.500 + 0.162i)13-s + (0.232 − 0.168i)14-s + (0.125 + 0.0914i)15-s + (0.283 + 0.872i)16-s + (−1.31 + 0.428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.63956 - 1.08930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63956 - 1.08930i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.54e3 - 1.45e4i)T \) |
| good | 2 | \( 1 + (-2.53 - 0.824i)T + (207. + 150. i)T^{2} \) |
| 3 | \( 1 + (-83.6 + 60.7i)T + (2.02e3 - 6.23e3i)T^{2} \) |
| 5 | \( 1 + (-23.5 - 72.4i)T + (-3.16e5 + 2.29e5i)T^{2} \) |
| 7 | \( 1 + (-2.43e3 + 3.34e3i)T + (-1.78e6 - 5.48e6i)T^{2} \) |
| 13 | \( 1 + (-1.42e4 - 4.64e3i)T + (6.59e8 + 4.79e8i)T^{2} \) |
| 17 | \( 1 + (1.10e5 - 3.57e4i)T + (5.64e9 - 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-4.87e4 - 6.71e4i)T + (-5.24e9 + 1.61e10i)T^{2} \) |
| 23 | \( 1 - 2.69e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (1.78e5 - 2.45e5i)T + (-1.54e11 - 4.75e11i)T^{2} \) |
| 31 | \( 1 + (-2.13e5 + 6.58e5i)T + (-6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (1.40e6 + 1.02e6i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (1.00e6 + 1.38e6i)T + (-2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 - 1.62e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-1.73e6 + 1.25e6i)T + (7.35e12 - 2.26e13i)T^{2} \) |
| 53 | \( 1 + (3.82e6 - 1.17e7i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-1.21e7 - 8.86e6i)T + (4.53e13 + 1.39e14i)T^{2} \) |
| 61 | \( 1 + (3.05e6 - 9.91e5i)T + (1.55e14 - 1.12e14i)T^{2} \) |
| 67 | \( 1 + 3.05e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (5.51e6 + 1.69e7i)T + (-5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.82e7 - 2.50e7i)T + (-2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (3.68e7 + 1.19e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (8.27e7 - 2.68e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 - 8.06e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-2.45e7 + 7.54e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−18.45208731234722934323018986083, −17.38513172148530672826989916827, −14.88340318949407083582741623506, −14.00423244433190216035560429703, −13.10090718770688847991319664972, −10.61551637608257059689482150070, −8.739496593974122729901242325196, −7.20866174778735272310718577657, −4.36671700333722741230608050117, −1.45266223200001940843973521344,
3.02886862264430435121459760474, 4.94374717068204730635255592613, 8.585535225461988679871207522124, 8.951163973219459949469529644639, 11.48633134256714637702286105139, 13.35882407670534636662447200409, 14.60248075431359578315435685050, 15.73914635942466413783816027785, 17.65616957635289257134871085822, 18.85412597985614184009277376745
