| L(s) = 1 | + (0.747 + 0.542i)2-s + (−0.476 + 1.46i)3-s + (−2.20 − 6.79i)4-s + (−7.05 + 5.12i)5-s + (−1.15 + 0.836i)6-s + (0.239 + 0.737i)7-s + (4.32 − 13.3i)8-s + (19.9 + 14.4i)9-s − 8.05·10-s + (32.3 + 16.8i)11-s + 11.0·12-s + (−52.2 − 37.9i)13-s + (−0.221 + 0.680i)14-s + (−4.15 − 12.7i)15-s + (−35.8 + 26.0i)16-s + (−56.7 + 41.2i)17-s + ⋯ |
| L(s) = 1 | + (0.264 + 0.191i)2-s + (−0.0916 + 0.282i)3-s + (−0.276 − 0.849i)4-s + (−0.631 + 0.458i)5-s + (−0.0783 + 0.0569i)6-s + (0.0129 + 0.0398i)7-s + (0.191 − 0.587i)8-s + (0.737 + 0.536i)9-s − 0.254·10-s + (0.887 + 0.461i)11-s + 0.265·12-s + (−1.11 − 0.810i)13-s + (−0.00422 + 0.0129i)14-s + (−0.0715 − 0.220i)15-s + (−0.559 + 0.406i)16-s + (−0.809 + 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.903164 + 0.0781412i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.903164 + 0.0781412i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-32.3 - 16.8i)T \) |
| good | 2 | \( 1 + (-0.747 - 0.542i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (0.476 - 1.46i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (7.05 - 5.12i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (-0.239 - 0.737i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (52.2 + 37.9i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (56.7 - 41.2i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-40.0 + 123. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (21.2 + 65.3i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (4.63 + 3.36i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-53.2 - 163. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (65.3 - 201. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (59.0 - 181. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (519. + 377. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-16.0 - 49.4i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-398. + 289. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-147. + 107. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (64.2 + 197. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (528. + 383. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (972. - 706. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 716.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-508. - 369. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−19.86766789331414677345136192476, −19.10500498353451245877147969178, −17.47381591369639311135036267627, −15.53938036788274268438005076924, −14.86812350942138223394832950016, −13.11887775095989291991778399578, −11.08609926178566813637797037704, −9.628690384199445352576498222380, −7.06821424214995517744993696332, −4.70378310900150901090610350822,
4.17494765173152190208858738824, 7.29949774709385328752952830106, 9.103108914830104546418463265988, 11.75215950459017834414429833599, 12.55823521839614835080001166371, 14.20689730368077320626472768149, 16.15012793727240559641515813285, 17.28761859333008092246271511018, 18.82187852890456561081254530695, 20.25674848036352470895890250638
