| L(s) = 1 | + (−10.8 − 33.3i)2-s + (−181. − 132. i)3-s + (−582. + 423. i)4-s + (311. − 957. i)5-s + (−2.43e3 + 7.50e3i)6-s + (7.26e3 − 5.28e3i)7-s + (5.90e3 + 4.28e3i)8-s + (9.54e3 + 2.93e4i)9-s − 3.53e4·10-s + (−3.85e4 + 2.95e4i)11-s + 1.61e5·12-s + (−6.77e3 − 2.08e4i)13-s + (−2.55e5 − 1.85e5i)14-s + (−1.83e5 + 1.33e5i)15-s + (−3.47e4 + 1.06e5i)16-s + (1.76e5 − 5.43e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.479 − 1.47i)2-s + (−1.29 − 0.942i)3-s + (−1.13 + 0.826i)4-s + (0.222 − 0.685i)5-s + (−0.768 + 2.36i)6-s + (1.14 − 0.831i)7-s + (0.509 + 0.370i)8-s + (0.484 + 1.49i)9-s − 1.11·10-s + (−0.793 + 0.608i)11-s + 2.25·12-s + (−0.0658 − 0.202i)13-s + (−1.77 − 1.28i)14-s + (−0.934 + 0.678i)15-s + (−0.132 + 0.408i)16-s + (0.512 − 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0200 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0200 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.407123 + 0.399059i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.407123 + 0.399059i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (3.85e4 - 2.95e4i)T \) |
| good | 2 | \( 1 + (10.8 + 33.3i)T + (-414. + 300. i)T^{2} \) |
| 3 | \( 1 + (181. + 132. i)T + (6.08e3 + 1.87e4i)T^{2} \) |
| 5 | \( 1 + (-311. + 957. i)T + (-1.58e6 - 1.14e6i)T^{2} \) |
| 7 | \( 1 + (-7.26e3 + 5.28e3i)T + (1.24e7 - 3.83e7i)T^{2} \) |
| 13 | \( 1 + (6.77e3 + 2.08e4i)T + (-8.57e9 + 6.23e9i)T^{2} \) |
| 17 | \( 1 + (-1.76e5 + 5.43e5i)T + (-9.59e10 - 6.97e10i)T^{2} \) |
| 19 | \( 1 + (-3.97e5 - 2.88e5i)T + (9.97e10 + 3.06e11i)T^{2} \) |
| 23 | \( 1 + 1.91e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (-1.59e5 + 1.15e5i)T + (4.48e12 - 1.37e13i)T^{2} \) |
| 31 | \( 1 + (6.98e5 + 2.14e6i)T + (-2.13e13 + 1.55e13i)T^{2} \) |
| 37 | \( 1 + (-3.94e6 + 2.86e6i)T + (4.01e13 - 1.23e14i)T^{2} \) |
| 41 | \( 1 + (5.49e6 + 3.99e6i)T + (1.01e14 + 3.11e14i)T^{2} \) |
| 43 | \( 1 + 3.04e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.13e7 + 1.55e7i)T + (3.45e14 + 1.06e15i)T^{2} \) |
| 53 | \( 1 + (-8.53e6 - 2.62e7i)T + (-2.66e15 + 1.93e15i)T^{2} \) |
| 59 | \( 1 + (-6.55e7 + 4.76e7i)T + (2.67e15 - 8.23e15i)T^{2} \) |
| 61 | \( 1 + (1.28e7 - 3.95e7i)T + (-9.46e15 - 6.87e15i)T^{2} \) |
| 67 | \( 1 + 1.82e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-8.49e7 + 2.61e8i)T + (-3.70e16 - 2.69e16i)T^{2} \) |
| 73 | \( 1 + (-1.34e8 + 9.73e7i)T + (1.81e16 - 5.59e16i)T^{2} \) |
| 79 | \( 1 + (-2.01e7 - 6.18e7i)T + (-9.69e16 + 7.04e16i)T^{2} \) |
| 83 | \( 1 + (3.39e7 - 1.04e8i)T + (-1.51e17 - 1.09e17i)T^{2} \) |
| 89 | \( 1 - 4.76e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-4.77e7 - 1.46e8i)T + (-6.15e17 + 4.46e17i)T^{2} \) |
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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
−17.94361778323462827657033532049, −16.64678683110345639254762697157, −13.53537427631477244556435827926, −12.26544871509407052746960760635, −11.41117522837235724675813199284, −10.08439412077636722250879603313, −7.68074198685421249513991387049, −5.04316233645854968463941002543, −1.63993538664912316062977952344, −0.48364672439923201747227833535,
5.16006311871390757890251835492, 6.15540183112466127148341356114, 8.257470606100872415695985934980, 10.22364752804746828288088364733, 11.60055075326504441689556322946, 14.51042418668015702237646731355, 15.49361506703484616201886738609, 16.53541060016509818087434758081, 17.74513447008871323514551713812, 18.36582228998164286226861607031
