| L(s) = 1 | + (−29.9 + 29.9i)2-s + (97.1 − 234. i)3-s − 1.28e3i·4-s + (−584. − 242. i)5-s + (4.11e3 + 9.93e3i)6-s + (−4.41e3 + 1.82e3i)7-s + (2.30e4 + 2.30e4i)8-s + (−3.16e4 − 3.16e4i)9-s + (2.47e4 − 1.02e4i)10-s + (9.13e3 + 2.20e4i)11-s + (−3.00e5 − 1.24e5i)12-s + 1.26e5i·13-s + (7.74e4 − 1.86e5i)14-s + (−1.13e5 + 1.13e5i)15-s − 7.24e5·16-s + (4.15e4 − 3.41e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.32 + 1.32i)2-s + (0.692 − 1.67i)3-s − 2.50i·4-s + (−0.418 − 0.173i)5-s + (1.29 + 3.12i)6-s + (−0.694 + 0.287i)7-s + (1.99 + 1.99i)8-s + (−1.60 − 1.60i)9-s + (0.783 − 0.324i)10-s + (0.188 + 0.453i)11-s + (−4.18 − 1.73i)12-s + 1.22i·13-s + (0.538 − 1.30i)14-s + (−0.579 + 0.579i)15-s − 2.76·16-s + (0.120 − 0.992i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0449i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.000733613 + 0.0326174i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000733613 + 0.0326174i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-4.15e4 + 3.41e5i)T \) |
| good | 2 | \( 1 + (29.9 - 29.9i)T - 512iT^{2} \) |
| 3 | \( 1 + (-97.1 + 234. i)T + (-1.39e4 - 1.39e4i)T^{2} \) |
| 5 | \( 1 + (584. + 242. i)T + (1.38e6 + 1.38e6i)T^{2} \) |
| 7 | \( 1 + (4.41e3 - 1.82e3i)T + (2.85e7 - 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-9.13e3 - 2.20e4i)T + (-1.66e9 + 1.66e9i)T^{2} \) |
| 13 | \( 1 - 1.26e5iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (6.00e5 - 6.00e5i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (-3.04e5 - 7.34e5i)T + (-1.27e12 + 1.27e12i)T^{2} \) |
| 29 | \( 1 + (7.98e5 + 3.30e5i)T + (1.02e13 + 1.02e13i)T^{2} \) |
| 31 | \( 1 + (1.94e5 - 4.70e5i)T + (-1.86e13 - 1.86e13i)T^{2} \) |
| 37 | \( 1 + (3.60e6 - 8.69e6i)T + (-9.18e13 - 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-1.99e6 + 8.28e5i)T + (2.31e14 - 2.31e14i)T^{2} \) |
| 43 | \( 1 + (9.48e6 + 9.48e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + 3.55e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (2.03e7 - 2.03e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (5.63e7 + 5.63e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (-6.43e7 + 2.66e7i)T + (8.26e15 - 8.26e15i)T^{2} \) |
| 67 | \( 1 + 1.51e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (3.46e7 - 8.37e7i)T + (-3.24e16 - 3.24e16i)T^{2} \) |
| 73 | \( 1 + (6.87e7 + 2.84e7i)T + (4.16e16 + 4.16e16i)T^{2} \) |
| 79 | \( 1 + (1.22e8 + 2.95e8i)T + (-8.47e16 + 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-4.89e8 + 4.89e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 - 6.16e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (6.65e8 + 2.75e8i)T + (5.37e17 + 5.37e17i)T^{2} \) |
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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
−16.31491339980228625920045969012, −14.87680332822664167048602181179, −13.75348453656185220248830522910, −12.03919968124578334654345158508, −9.465931181853032398469691783053, −8.370452733114205534037988820634, −7.22067222276457589334119312188, −6.31316573143454548198379827738, −1.73172100819441747890899121587, −0.02233238844815271164989508478,
2.90337107155231344735302948059, 3.90203351993852699867572859187, 8.151175075859278464224363916580, 9.208492087344991290508247441027, 10.36517265989294568879215317005, 11.02551576965108353827012250878, 12.99144992680138726147947550672, 15.14500699126963001913377482994, 16.34013384527788027293158815090, 17.38391871986732617178454247881
