| L(s) = 1 | + (−25.0 − 25.0i)2-s + (−61.4 − 148. i)3-s + 742. i·4-s + (−1.19e3 + 494. i)5-s + (−2.17e3 + 5.25e3i)6-s + (−4.19e3 − 1.73e3i)7-s + (5.76e3 − 5.76e3i)8-s + (−4.30e3 + 4.30e3i)9-s + (4.23e4 + 1.75e4i)10-s + (1.43e4 − 3.46e4i)11-s + (1.10e5 − 4.55e4i)12-s + 1.45e4i·13-s + (6.15e4 + 1.48e5i)14-s + (1.46e5 + 1.46e5i)15-s + 9.13e4·16-s + (−1.09e5 − 3.26e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.10 − 1.10i)2-s + (−0.437 − 1.05i)3-s + 1.44i·4-s + (−0.854 + 0.354i)5-s + (−0.685 + 1.65i)6-s + (−0.659 − 0.273i)7-s + (0.497 − 0.497i)8-s + (−0.218 + 0.218i)9-s + (1.33 + 0.554i)10-s + (0.295 − 0.714i)11-s + (1.53 − 0.634i)12-s + 0.140i·13-s + (0.427 + 1.03i)14-s + (0.748 + 0.748i)15-s + 0.348·16-s + (−0.317 − 0.948i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.129730 + 0.0161198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.129730 + 0.0161198i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (1.09e5 + 3.26e5i)T \) |
| good | 2 | \( 1 + (25.0 + 25.0i)T + 512iT^{2} \) |
| 3 | \( 1 + (61.4 + 148. i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (1.19e3 - 494. i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (4.19e3 + 1.73e3i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-1.43e4 + 3.46e4i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 - 1.45e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-5.36e4 - 5.36e4i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (7.03e5 - 1.69e6i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-5.83e6 + 2.41e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-3.72e6 - 8.99e6i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-7.34e6 - 1.77e7i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (8.29e6 + 3.43e6i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (9.01e6 - 9.01e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 9.01e6iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (5.33e7 + 5.33e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (3.04e7 - 3.04e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (1.30e7 + 5.41e6i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 + 9.01e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-7.82e7 - 1.88e8i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-7.43e7 + 3.08e7i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (1.52e8 - 3.68e8i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-1.31e8 - 1.31e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 5.13e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-6.55e8 + 2.71e8i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−17.38690887031690260904588438599, −15.89105902580880537772854643463, −13.57451605700345314067337581958, −12.02906369535266624482395895580, −11.45156957374125154042063732060, −9.847636037005427601545828954433, −8.129981395672991658434996470458, −6.72309978934648411575006644200, −3.20635976030098056949753600853, −1.12818334070159162777975045935,
0.11888501206121676187813729508, 4.33947622729911546940402808118, 6.27807386991897037950474157530, 8.002508746014106691005550859411, 9.371524366181968594367233399282, 10.48463125666718767066454048453, 12.36402164285460451735786588473, 14.95981225301080929035885073395, 15.78989439567772402852876005757, 16.44892827025304291772676069650
