| L(s) = 1 | + (−43.8 + 60.4i)2-s + (−68.3 + 210. i)3-s + (−457. − 1.40e3i)4-s + (−1.15e4 + 8.37e3i)5-s + (−9.71e3 − 1.33e4i)6-s + (−1.04e5 + 3.40e4i)7-s + (−1.85e5 − 6.03e4i)8-s + (3.90e5 + 2.83e5i)9-s − 1.06e6i·10-s + (9.88e5 − 1.46e6i)11-s + 3.27e5·12-s + (2.00e6 − 2.75e6i)13-s + (2.54e6 − 7.82e6i)14-s + (−9.74e5 − 2.99e6i)15-s + (1.67e7 − 1.21e7i)16-s + (2.67e6 + 3.68e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.685 + 0.944i)2-s + (−0.0938 + 0.288i)3-s + (−0.111 − 0.343i)4-s + (−0.737 + 0.536i)5-s + (−0.208 − 0.286i)6-s + (−0.890 + 0.289i)7-s + (−0.708 − 0.230i)8-s + (0.734 + 0.533i)9-s − 1.06i·10-s + (0.558 − 0.829i)11-s + 0.109·12-s + (0.415 − 0.571i)13-s + (0.337 − 1.03i)14-s + (−0.0855 − 0.263i)15-s + (0.995 − 0.723i)16-s + (0.111 + 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0231029 - 0.0173804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0231029 - 0.0173804i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-9.88e5 + 1.46e6i)T \) |
| good | 2 | \( 1 + (43.8 - 60.4i)T + (-1.26e3 - 3.89e3i)T^{2} \) |
| 3 | \( 1 + (68.3 - 210. i)T + (-4.29e5 - 3.12e5i)T^{2} \) |
| 5 | \( 1 + (1.15e4 - 8.37e3i)T + (7.54e7 - 2.32e8i)T^{2} \) |
| 7 | \( 1 + (1.04e5 - 3.40e4i)T + (1.11e10 - 8.13e9i)T^{2} \) |
| 13 | \( 1 + (-2.00e6 + 2.75e6i)T + (-7.19e12 - 2.21e13i)T^{2} \) |
| 17 | \( 1 + (-2.67e6 - 3.68e6i)T + (-1.80e14 + 5.54e14i)T^{2} \) |
| 19 | \( 1 + (5.66e7 + 1.84e7i)T + (1.79e15 + 1.30e15i)T^{2} \) |
| 23 | \( 1 - 5.88e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (9.20e7 - 2.99e7i)T + (2.86e17 - 2.07e17i)T^{2} \) |
| 31 | \( 1 + (1.26e9 + 9.18e8i)T + (2.43e17 + 7.49e17i)T^{2} \) |
| 37 | \( 1 + (5.73e8 + 1.76e9i)T + (-5.32e18 + 3.86e18i)T^{2} \) |
| 41 | \( 1 + (1.75e9 + 5.70e8i)T + (1.82e19 + 1.32e19i)T^{2} \) |
| 43 | \( 1 - 1.26e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (4.94e9 - 1.52e10i)T + (-9.40e19 - 6.82e19i)T^{2} \) |
| 53 | \( 1 + (-1.31e10 - 9.56e9i)T + (1.51e20 + 4.67e20i)T^{2} \) |
| 59 | \( 1 + (1.66e9 + 5.13e9i)T + (-1.43e21 + 1.04e21i)T^{2} \) |
| 61 | \( 1 + (5.87e10 + 8.09e10i)T + (-8.20e20 + 2.52e21i)T^{2} \) |
| 67 | \( 1 + 1.35e11T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-2.01e11 + 1.46e11i)T + (5.07e21 - 1.56e22i)T^{2} \) |
| 73 | \( 1 + (1.09e11 - 3.54e10i)T + (1.85e22 - 1.34e22i)T^{2} \) |
| 79 | \( 1 + (-9.19e10 + 1.26e11i)T + (-1.82e22 - 5.61e22i)T^{2} \) |
| 83 | \( 1 + (1.05e11 + 1.44e11i)T + (-3.30e22 + 1.01e23i)T^{2} \) |
| 89 | \( 1 + 1.89e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-4.62e11 - 3.35e11i)T + (2.14e23 + 6.59e23i)T^{2} \) |
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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
−18.59672833644468508123046390396, −16.88547436013777024026756745696, −15.92096801740324827575399184691, −15.04354128839670469515148702383, −12.84409871376669384207089321873, −10.93475208660737340906485212618, −9.160208065789249369508976710096, −7.61982527789651249736146678267, −6.22199232093475302266755752630, −3.49635214375994783402398003296,
0.01778764821242994608992567555, 1.51168087132057094938616088644, 3.86294203692796709571255680762, 6.75780907399467257039088405755, 8.900481039498299480122017324016, 10.17659678416760243213978922487, 11.88390135102486402652133556898, 12.78145173628766871442627420220, 15.10511718126346257564257824368, 16.61441664482539009941285511853
