| L(s) = 1 | + (7.79 + 5.66i)2-s + (51.4 − 158. i)3-s + (−129. − 398. i)4-s + (−1.98e3 + 1.44e3i)5-s + (1.29e3 − 943. i)6-s + (−2.29e3 − 7.07e3i)7-s + (2.77e3 − 8.52e3i)8-s + (−6.53e3 − 4.74e3i)9-s − 2.36e4·10-s + (4.54e4 − 1.71e4i)11-s − 6.98e4·12-s + (−2.97e4 − 2.16e4i)13-s + (2.21e4 − 6.81e4i)14-s + (1.26e5 + 3.88e5i)15-s + (−1.03e5 + 7.54e4i)16-s + (4.61e5 − 3.35e5i)17-s + ⋯ |
| L(s) = 1 | + (0.344 + 0.250i)2-s + (0.366 − 1.12i)3-s + (−0.253 − 0.778i)4-s + (−1.42 + 1.03i)5-s + (0.408 − 0.297i)6-s + (−0.361 − 1.11i)7-s + (0.239 − 0.736i)8-s + (−0.331 − 0.241i)9-s − 0.747·10-s + (0.935 − 0.353i)11-s − 0.972·12-s + (−0.288 − 0.209i)13-s + (0.154 − 0.474i)14-s + (0.644 + 1.98i)15-s + (−0.395 + 0.287i)16-s + (1.34 − 0.974i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.651508 - 1.17346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.651508 - 1.17346i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.54e4 + 1.71e4i)T \) |
| good | 2 | \( 1 + (-7.79 - 5.66i)T + (158. + 486. i)T^{2} \) |
| 3 | \( 1 + (-51.4 + 158. i)T + (-1.59e4 - 1.15e4i)T^{2} \) |
| 5 | \( 1 + (1.98e3 - 1.44e3i)T + (6.03e5 - 1.85e6i)T^{2} \) |
| 7 | \( 1 + (2.29e3 + 7.07e3i)T + (-3.26e7 + 2.37e7i)T^{2} \) |
| 13 | \( 1 + (2.97e4 + 2.16e4i)T + (3.27e9 + 1.00e10i)T^{2} \) |
| 17 | \( 1 + (-4.61e5 + 3.35e5i)T + (3.66e10 - 1.12e11i)T^{2} \) |
| 19 | \( 1 + (1.45e5 - 4.47e5i)T + (-2.61e11 - 1.89e11i)T^{2} \) |
| 23 | \( 1 + 9.05e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + (-2.80e5 - 8.61e5i)T + (-1.17e13 + 8.52e12i)T^{2} \) |
| 31 | \( 1 + (-2.01e6 - 1.46e6i)T + (8.17e12 + 2.51e13i)T^{2} \) |
| 37 | \( 1 + (2.41e6 + 7.42e6i)T + (-1.05e14 + 7.63e13i)T^{2} \) |
| 41 | \( 1 + (-2.34e6 + 7.21e6i)T + (-2.64e14 - 1.92e14i)T^{2} \) |
| 43 | \( 1 + 7.64e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-5.33e6 + 1.64e7i)T + (-9.05e14 - 6.57e14i)T^{2} \) |
| 53 | \( 1 + (2.20e7 + 1.59e7i)T + (1.01e15 + 3.13e15i)T^{2} \) |
| 59 | \( 1 + (-4.02e7 - 1.23e8i)T + (-7.00e15 + 5.09e15i)T^{2} \) |
| 61 | \( 1 + (-3.94e7 + 2.86e7i)T + (3.61e15 - 1.11e16i)T^{2} \) |
| 67 | \( 1 + 1.20e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-1.28e8 + 9.30e7i)T + (1.41e16 - 4.36e16i)T^{2} \) |
| 73 | \( 1 + (-1.53e7 - 4.70e7i)T + (-4.76e16 + 3.46e16i)T^{2} \) |
| 79 | \( 1 + (-4.86e8 - 3.53e8i)T + (3.70e16 + 1.13e17i)T^{2} \) |
| 83 | \( 1 + (-1.84e8 + 1.33e8i)T + (5.77e16 - 1.77e17i)T^{2} \) |
| 89 | \( 1 - 2.27e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-7.07e8 - 5.13e8i)T + (2.34e17 + 7.23e17i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.42845351487823737565902014118, −16.22488842291214121871111117365, −14.57555069368055497964413808118, −13.87137727400404390866314838638, −12.08075453515439506319010890822, −10.34518285899193337624699022323, −7.65332429024817129628553714629, −6.71492849413789841229029983393, −3.73102759691183503619490492995, −0.73719969176474858735057670488,
3.55001329964293507761153219608, 4.62971624044359518273643283605, 8.222533099642795306308772987033, 9.296369082576656104492500738188, 11.82166105811611478002505343164, 12.53994807437676493525625917056, 14.85778060855028104455680966802, 15.91238731462687806263057781772, 16.95025217050455027043563866848, 19.26552069285713469109227295223
