| L(s) = 1 | + (−67.5 − 92.9i)2-s + (−392. − 1.20e3i)3-s + (−2.81e3 + 8.66e3i)4-s + (1.66e4 + 1.20e4i)5-s + (−8.57e4 + 1.17e5i)6-s + (1.88e4 + 6.12e3i)7-s + (5.48e5 − 1.78e5i)8-s + (−8.72e5 + 6.34e5i)9-s − 2.36e6i·10-s + (−1.45e6 + 1.01e6i)11-s + 1.15e7·12-s + (2.46e6 + 3.39e6i)13-s + (−7.04e5 − 2.16e6i)14-s + (8.07e6 − 2.48e7i)15-s + (−2.33e7 − 1.69e7i)16-s + (−1.61e7 + 2.22e7i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 1.45i)2-s + (−0.537 − 1.65i)3-s + (−0.687 + 2.11i)4-s + (1.06 + 0.774i)5-s + (−1.83 + 2.52i)6-s + (0.160 + 0.0520i)7-s + (2.09 − 0.679i)8-s + (−1.64 + 1.19i)9-s − 2.36i·10-s + (−0.821 + 0.570i)11-s + 3.87·12-s + (0.510 + 0.703i)13-s + (−0.0935 − 0.287i)14-s + (0.708 − 2.18i)15-s + (−1.39 − 1.01i)16-s + (−0.669 + 0.921i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.452210 - 0.0604368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452210 - 0.0604368i\) |
| \(L(7)\) |
|
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 | \( 1 + (67.5 + 92.9i)T + (-1.26e3 + 3.89e3i)T^{2} \) |
| 3 | \( 1 + (392. + 1.20e3i)T + (-4.29e5 + 3.12e5i)T^{2} \) |
| 5 | \( 1 + (-1.66e4 - 1.20e4i)T + (7.54e7 + 2.32e8i)T^{2} \) |
| 7 | \( 1 + (-1.88e4 - 6.12e3i)T + (1.11e10 + 8.13e9i)T^{2} \) |
| 13 | \( 1 + (-2.46e6 - 3.39e6i)T + (-7.19e12 + 2.21e13i)T^{2} \) |
| 17 | \( 1 + (1.61e7 - 2.22e7i)T + (-1.80e14 - 5.54e14i)T^{2} \) |
| 19 | \( 1 + (6.15e6 - 1.99e6i)T + (1.79e15 - 1.30e15i)T^{2} \) |
| 23 | \( 1 + 1.54e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + (-3.19e8 - 1.03e8i)T + (2.86e17 + 2.07e17i)T^{2} \) |
| 31 | \( 1 + (-2.71e8 + 1.97e8i)T + (2.43e17 - 7.49e17i)T^{2} \) |
| 37 | \( 1 + (8.17e8 - 2.51e9i)T + (-5.32e18 - 3.86e18i)T^{2} \) |
| 41 | \( 1 + (2.63e9 - 8.55e8i)T + (1.82e19 - 1.32e19i)T^{2} \) |
| 43 | \( 1 - 3.92e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (-8.25e8 - 2.54e9i)T + (-9.40e19 + 6.82e19i)T^{2} \) |
| 53 | \( 1 + (-1.96e9 + 1.42e9i)T + (1.51e20 - 4.67e20i)T^{2} \) |
| 59 | \( 1 + (-1.83e10 + 5.63e10i)T + (-1.43e21 - 1.04e21i)T^{2} \) |
| 61 | \( 1 + (5.87e10 - 8.09e10i)T + (-8.20e20 - 2.52e21i)T^{2} \) |
| 67 | \( 1 - 1.44e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (9.47e10 + 6.88e10i)T + (5.07e21 + 1.56e22i)T^{2} \) |
| 73 | \( 1 + (-1.65e11 - 5.37e10i)T + (1.85e22 + 1.34e22i)T^{2} \) |
| 79 | \( 1 + (-7.43e10 - 1.02e11i)T + (-1.82e22 + 5.61e22i)T^{2} \) |
| 83 | \( 1 + (2.11e11 - 2.91e11i)T + (-3.30e22 - 1.01e23i)T^{2} \) |
| 89 | \( 1 + 6.38e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-1.00e12 + 7.27e11i)T + (2.14e23 - 6.59e23i)T^{2} \) |
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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
−18.04889922355271256563171495244, −17.21052586930935157239523409092, −13.70865231717883348159712722111, −12.68669924684255989862592524070, −11.38668583102311136252856416426, −10.17864100769337770637917395943, −8.192693049035693068319929129975, −6.49199585247230433785386533894, −2.36135243948127442016488131437, −1.56996819910507605642480635975,
0.32452955623280107640825990652, 5.02996447549194561790747025162, 5.89844211529596419105365469594, 8.535815624041942750690828503181, 9.622792248770334830009516844649, 10.63257167626950930996234418608, 13.89658296129912152421171733995, 15.57175890476317119107562424902, 16.16017666264787967559180006192, 17.23672025339166973037059241443
