| L(s) = 1 | + (59.8 − 82.4i)2-s + (267. − 822. i)3-s + (−1.94e3 − 5.97e3i)4-s + (5.39e3 − 3.92e3i)5-s + (−5.17e4 − 7.12e4i)6-s + (477. − 155. i)7-s + (−2.11e5 − 6.88e4i)8-s + (−1.75e5 − 1.27e5i)9-s − 6.79e5i·10-s + (1.45e6 + 1.01e6i)11-s − 5.43e6·12-s + (−3.35e6 + 4.62e6i)13-s + (1.58e4 − 4.86e4i)14-s + (−1.78e6 − 5.48e6i)15-s + (2.46e6 − 1.79e6i)16-s + (−1.38e7 − 1.90e7i)17-s + ⋯ |
| L(s) = 1 | + (0.935 − 1.28i)2-s + (0.366 − 1.12i)3-s + (−0.473 − 1.45i)4-s + (0.345 − 0.250i)5-s + (−1.11 − 1.52i)6-s + (0.00405 − 0.00131i)7-s + (−0.807 − 0.262i)8-s + (−0.330 − 0.239i)9-s − 0.679i·10-s + (0.819 + 0.572i)11-s − 1.81·12-s + (−0.695 + 0.957i)13-s + (0.00209 − 0.00646i)14-s + (−0.156 − 0.481i)15-s + (0.147 − 0.106i)16-s + (−0.573 − 0.789i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.428257 - 3.23200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428257 - 3.23200i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-1.45e6 - 1.01e6i)T \) |
| good | 2 | \( 1 + (-59.8 + 82.4i)T + (-1.26e3 - 3.89e3i)T^{2} \) |
| 3 | \( 1 + (-267. + 822. i)T + (-4.29e5 - 3.12e5i)T^{2} \) |
| 5 | \( 1 + (-5.39e3 + 3.92e3i)T + (7.54e7 - 2.32e8i)T^{2} \) |
| 7 | \( 1 + (-477. + 155. i)T + (1.11e10 - 8.13e9i)T^{2} \) |
| 13 | \( 1 + (3.35e6 - 4.62e6i)T + (-7.19e12 - 2.21e13i)T^{2} \) |
| 17 | \( 1 + (1.38e7 + 1.90e7i)T + (-1.80e14 + 5.54e14i)T^{2} \) |
| 19 | \( 1 + (3.39e7 + 1.10e7i)T + (1.79e15 + 1.30e15i)T^{2} \) |
| 23 | \( 1 - 2.61e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-5.80e8 + 1.88e8i)T + (2.86e17 - 2.07e17i)T^{2} \) |
| 31 | \( 1 + (9.03e8 + 6.56e8i)T + (2.43e17 + 7.49e17i)T^{2} \) |
| 37 | \( 1 + (-3.64e8 - 1.12e9i)T + (-5.32e18 + 3.86e18i)T^{2} \) |
| 41 | \( 1 + (1.73e9 + 5.64e8i)T + (1.82e19 + 1.32e19i)T^{2} \) |
| 43 | \( 1 + 1.40e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (4.86e9 - 1.49e10i)T + (-9.40e19 - 6.82e19i)T^{2} \) |
| 53 | \( 1 + (-6.76e9 - 4.91e9i)T + (1.51e20 + 4.67e20i)T^{2} \) |
| 59 | \( 1 + (-9.91e9 - 3.05e10i)T + (-1.43e21 + 1.04e21i)T^{2} \) |
| 61 | \( 1 + (1.92e8 + 2.64e8i)T + (-8.20e20 + 2.52e21i)T^{2} \) |
| 67 | \( 1 - 1.65e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (1.05e11 - 7.69e10i)T + (5.07e21 - 1.56e22i)T^{2} \) |
| 73 | \( 1 + (9.95e10 - 3.23e10i)T + (1.85e22 - 1.34e22i)T^{2} \) |
| 79 | \( 1 + (4.03e10 - 5.55e10i)T + (-1.82e22 - 5.61e22i)T^{2} \) |
| 83 | \( 1 + (-2.51e11 - 3.45e11i)T + (-3.30e22 + 1.01e23i)T^{2} \) |
| 89 | \( 1 - 1.39e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (2.28e11 + 1.66e11i)T + (2.14e23 + 6.59e23i)T^{2} \) |
| 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
−17.20746377788832729554704605104, −14.60555819749973636477671347721, −13.47308414659045110651936424290, −12.59314337702440203505561402324, −11.39762328922302789910303677425, −9.382121635098394367080463557500, −6.96336399177977719650146061987, −4.61236950148098986770659024261, −2.45812439553464248237262516971, −1.32935932956989060985093869212,
3.52494470736067819587802959663, 4.97795039544189408572620777432, 6.61258708052222144290764163989, 8.637436090346783675368192643451, 10.44251956212542570107702497124, 12.89978364217177490460015452392, 14.48471759437789285262247716269, 15.06726848845687900281700459921, 16.31236567935842451617997352794, 17.43193648179120803843399228333
