| L(s) = 1 | + (−12.7 − 4.15i)2-s + (0.366 − 0.266i)3-s + (94.5 + 68.7i)4-s + (50.5 + 155. i)5-s + (−5.79 + 1.88i)6-s + (50.6 − 69.7i)7-s + (−418. − 575. i)8-s + (−225. + 693. i)9-s − 2.20e3i·10-s + (−990. + 888. i)11-s + 52.9·12-s + (3.65e3 + 1.18e3i)13-s + (−938. + 681. i)14-s + (59.9 + 43.5i)15-s + (644. + 1.98e3i)16-s + (−5.29e3 + 1.71e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.519i)2-s + (0.0135 − 0.00985i)3-s + (1.47 + 1.07i)4-s + (0.404 + 1.24i)5-s + (−0.0268 + 0.00871i)6-s + (0.147 − 0.203i)7-s + (−0.816 − 1.12i)8-s + (−0.308 + 0.950i)9-s − 2.20i·10-s + (−0.744 + 0.667i)11-s + 0.0306·12-s + (1.66 + 0.540i)13-s + (−0.341 + 0.248i)14-s + (0.0177 + 0.0129i)15-s + (0.157 + 0.484i)16-s + (−1.07 + 0.350i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.528315 + 0.296666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528315 + 0.296666i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (990. - 888. i)T \) |
| good | 2 | \( 1 + (12.7 + 4.15i)T + (51.7 + 37.6i)T^{2} \) |
| 3 | \( 1 + (-0.366 + 0.266i)T + (225. - 693. i)T^{2} \) |
| 5 | \( 1 + (-50.5 - 155. i)T + (-1.26e4 + 9.18e3i)T^{2} \) |
| 7 | \( 1 + (-50.6 + 69.7i)T + (-3.63e4 - 1.11e5i)T^{2} \) |
| 13 | \( 1 + (-3.65e3 - 1.18e3i)T + (3.90e6 + 2.83e6i)T^{2} \) |
| 17 | \( 1 + (5.29e3 - 1.71e3i)T + (1.95e7 - 1.41e7i)T^{2} \) |
| 19 | \( 1 + (3.01e3 + 4.15e3i)T + (-1.45e7 + 4.47e7i)T^{2} \) |
| 23 | \( 1 - 1.08e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-7.98e3 + 1.09e4i)T + (-1.83e8 - 5.65e8i)T^{2} \) |
| 31 | \( 1 + (6.68e3 - 2.05e4i)T + (-7.18e8 - 5.21e8i)T^{2} \) |
| 37 | \( 1 + (-9.08e3 - 6.60e3i)T + (7.92e8 + 2.44e9i)T^{2} \) |
| 41 | \( 1 + (1.37e4 + 1.89e4i)T + (-1.46e9 + 4.51e9i)T^{2} \) |
| 43 | \( 1 + 9.41e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.51e5 + 1.09e5i)T + (3.33e9 - 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-1.54e4 + 4.75e4i)T + (-1.79e10 - 1.30e10i)T^{2} \) |
| 59 | \( 1 + (-8.38e3 - 6.09e3i)T + (1.30e10 + 4.01e10i)T^{2} \) |
| 61 | \( 1 + (6.26e4 - 2.03e4i)T + (4.16e10 - 3.02e10i)T^{2} \) |
| 67 | \( 1 - 3.66e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (1.74e4 + 5.38e4i)T + (-1.03e11 + 7.52e10i)T^{2} \) |
| 73 | \( 1 + (2.38e5 - 3.28e5i)T + (-4.67e10 - 1.43e11i)T^{2} \) |
| 79 | \( 1 + (-3.68e5 - 1.19e5i)T + (1.96e11 + 1.42e11i)T^{2} \) |
| 83 | \( 1 + (-2.91e4 + 9.48e3i)T + (2.64e11 - 1.92e11i)T^{2} \) |
| 89 | \( 1 - 2.79e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.12e5 - 3.46e5i)T + (-6.73e11 - 4.89e11i)T^{2} \) |
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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
−19.07481798344127691308371505916, −18.26775209153213773370992414341, −17.23807845560424572353855865132, −15.62966849357117357060217439453, −13.62611914167252011427780035740, −11.03409181877520848421078086331, −10.55155879432803521832079505483, −8.657992977301447873830626944131, −6.97892715798207790608207526352, −2.27869875275523118568740720533,
0.867952115564146671044065315942, 6.02453135631383266079634438249, 8.392006738251750077812371836960, 9.124776982617799076289295290046, 10.96578711751549903043182698386, 13.11162806728659382204635293752, 15.47104834193665302531077259615, 16.47540066015705038627292742631, 17.67022441460405632960841876659, 18.56390965279128346081857670654
