| L(s) = 1 | + (−14.5 + 14.5i)2-s + (32.3 − 78.1i)3-s + 88.7i·4-s + (31.3 + 12.9i)5-s + (665. + 1.60e3i)6-s + (−4.77e3 + 1.97e3i)7-s + (−8.73e3 − 8.73e3i)8-s + (8.86e3 + 8.86e3i)9-s + (−644. + 267. i)10-s + (−2.56e4 − 6.19e4i)11-s + (6.93e3 + 2.87e3i)12-s − 1.94e5i·13-s + (4.06e4 − 9.82e4i)14-s + (2.02e3 − 2.02e3i)15-s + 2.08e5·16-s + (−3.20e5 − 1.25e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.642 + 0.642i)2-s + (0.230 − 0.556i)3-s + 0.173i·4-s + (0.0224 + 0.00928i)5-s + (0.209 + 0.506i)6-s + (−0.751 + 0.311i)7-s + (−0.754 − 0.754i)8-s + (0.450 + 0.450i)9-s + (−0.0203 + 0.00844i)10-s + (−0.528 − 1.27i)11-s + (0.0965 + 0.0399i)12-s − 1.89i·13-s + (0.283 − 0.683i)14-s + (0.0103 − 0.0103i)15-s + 0.796·16-s + (−0.931 − 0.364i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.293 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.256328 - 0.346702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256328 - 0.346702i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (3.20e5 + 1.25e5i)T \) |
| good | 2 | \( 1 + (14.5 - 14.5i)T - 512iT^{2} \) |
| 3 | \( 1 + (-32.3 + 78.1i)T + (-1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-31.3 - 12.9i)T + (1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (4.77e3 - 1.97e3i)T + (2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (2.56e4 + 6.19e4i)T + (-1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 + 1.94e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (3.28e5 - 3.28e5i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (-1.44e5 - 3.48e5i)T + (-1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (9.41e5 + 3.89e5i)T + (1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (2.32e6 - 5.62e6i)T + (-1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (1.67e6 - 4.05e6i)T + (-9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-1.46e7 + 6.06e6i)T + (2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (3.45e6 + 3.45e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + 1.42e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (2.43e7 - 2.43e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (-7.93e7 - 7.93e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (1.65e8 - 6.85e7i)T + (8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 - 1.06e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-1.47e8 + 3.54e8i)T + (-3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (3.98e8 + 1.65e8i)T + (4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (8.17e7 + 1.97e8i)T + (-8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (2.68e8 - 2.68e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + 9.12e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-1.34e8 - 5.58e7i)T + (5.37e17 + 5.37e17i)T^{2} \) |
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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
−16.25586446976275301036828761196, −15.52613150673051166379169814844, −13.41837308502328428772116292171, −12.57266946004889943767084975054, −10.41076380698201367844013988363, −8.637031443826259675619498483165, −7.61754481613169706270077854509, −6.03570096782292375858797702309, −3.01858213806027625887541041511, −0.24364386873651023492834136828,
2.01831170050171971064370663679, 4.28456326950604221489207138881, 6.74421477645859756655216236817, 9.195906177241538239939352282170, 9.825881409589287513594996072993, 11.23295238879234540760809050150, 12.87904084782664194412068057774, 14.67606022776322747499830454910, 15.75346511720673872786839970896, 17.32812098358408635613680913184
