| L(s) = 1 | + (18.6 + 18.6i)2-s + (−274. − 663. i)3-s − 1.34e3i·4-s + (1.89e3 − 783. i)5-s + (7.26e3 − 1.75e4i)6-s + (−1.94e4 − 8.05e3i)7-s + (6.35e4 − 6.35e4i)8-s + (−2.39e5 + 2.39e5i)9-s + (5.00e4 + 2.07e4i)10-s + (−2.12e5 + 5.13e5i)11-s + (−8.95e5 + 3.70e5i)12-s − 7.21e4i·13-s + (−2.13e5 − 5.14e5i)14-s + (−1.03e6 − 1.03e6i)15-s − 3.88e5·16-s + (55.2 − 5.85e6i)17-s + ⋯ |
| L(s) = 1 | + (0.413 + 0.413i)2-s + (−0.652 − 1.57i)3-s − 0.658i·4-s + (0.270 − 0.112i)5-s + (0.381 − 0.920i)6-s + (−0.437 − 0.181i)7-s + (0.685 − 0.685i)8-s + (−1.35 + 1.35i)9-s + (0.158 + 0.0655i)10-s + (−0.398 + 0.961i)11-s + (−1.03 + 0.430i)12-s − 0.0539i·13-s + (−0.105 − 0.255i)14-s + (−0.353 − 0.353i)15-s − 0.0926·16-s + (9.43e−6 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0758i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0395133 + 1.03975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0395133 + 1.03975i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-55.2 + 5.85e6i)T \) |
| good | 2 | \( 1 + (-18.6 - 18.6i)T + 2.04e3iT^{2} \) |
| 3 | \( 1 + (274. + 663. i)T + (-1.25e5 + 1.25e5i)T^{2} \) |
| 5 | \( 1 + (-1.89e3 + 783. i)T + (3.45e7 - 3.45e7i)T^{2} \) |
| 7 | \( 1 + (1.94e4 + 8.05e3i)T + (1.39e9 + 1.39e9i)T^{2} \) |
| 11 | \( 1 + (2.12e5 - 5.13e5i)T + (-2.01e11 - 2.01e11i)T^{2} \) |
| 13 | \( 1 + 7.21e4iT - 1.79e12T^{2} \) |
| 19 | \( 1 + (8.42e5 + 8.42e5i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + (6.36e6 - 1.53e7i)T + (-6.73e14 - 6.73e14i)T^{2} \) |
| 29 | \( 1 + (-1.14e8 + 4.74e7i)T + (8.62e15 - 8.62e15i)T^{2} \) |
| 31 | \( 1 + (6.49e7 + 1.56e8i)T + (-1.79e16 + 1.79e16i)T^{2} \) |
| 37 | \( 1 + (-2.51e8 - 6.06e8i)T + (-1.25e17 + 1.25e17i)T^{2} \) |
| 41 | \( 1 + (1.08e9 + 4.47e8i)T + (3.89e17 + 3.89e17i)T^{2} \) |
| 43 | \( 1 + (-7.28e8 + 7.28e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.52e9iT - 2.47e18T^{2} \) |
| 53 | \( 1 + (6.14e7 + 6.14e7i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-5.34e9 + 5.34e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (8.61e9 + 3.56e9i)T + (3.07e19 + 3.07e19i)T^{2} \) |
| 67 | \( 1 - 9.16e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (7.19e9 + 1.73e10i)T + (-1.63e20 + 1.63e20i)T^{2} \) |
| 73 | \( 1 + (6.96e9 - 2.88e9i)T + (2.21e20 - 2.21e20i)T^{2} \) |
| 79 | \( 1 + (-5.71e8 + 1.38e9i)T + (-5.28e20 - 5.28e20i)T^{2} \) |
| 83 | \( 1 + (4.11e10 + 4.11e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 - 7.13e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + (1.07e11 - 4.46e10i)T + (5.05e21 - 5.05e21i)T^{2} \) |
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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
−15.46231840747114655590134218887, −13.78226013820324240423416312519, −13.08613090360479337036950690657, −11.71106245689108575070813402704, −9.937744144328234760183694810107, −7.46607450865417350851190921623, −6.46626061754489918142680643513, −5.18671651628577487271322447669, −1.89025526411099286678454561894, −0.40595011660416146554438920670,
3.07512005370217102347159550499, 4.34842658026822725533590912496, 5.89855448545637745890470616632, 8.603053285495085487771605128139, 10.24573782099057873898320153871, 11.21112805655233519260312807289, 12.60767823535101623440292056165, 14.25116929467510048186409152424, 15.88683361364560749025975972923, 16.57577734125648911426149811846
