| L(s) = 1 | + (−24.8 + 24.8i)2-s + (109. + 45.3i)3-s − 718. i·4-s + (−625. + 1.51e3i)5-s + (−3.84e3 + 1.59e3i)6-s + (4.12e3 + 9.96e3i)7-s + (5.11e3 + 5.11e3i)8-s + (−3.97e3 − 3.97e3i)9-s + (−2.19e4 − 5.29e4i)10-s + (−1.95e4 + 8.11e3i)11-s + (3.25e4 − 7.86e4i)12-s − 7.10e4i·13-s + (−3.49e5 − 1.44e5i)14-s + (−1.37e5 + 1.37e5i)15-s + 1.13e5·16-s + (4.41e4 − 3.41e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 + 1.09i)2-s + (0.780 + 0.323i)3-s − 1.40i·4-s + (−0.447 + 1.08i)5-s + (−1.21 + 0.501i)6-s + (0.649 + 1.56i)7-s + (0.441 + 0.441i)8-s + (−0.202 − 0.202i)9-s + (−0.694 − 1.67i)10-s + (−0.403 + 0.167i)11-s + (0.453 − 1.09i)12-s − 0.689i·13-s + (−2.43 − 1.00i)14-s + (−0.699 + 0.699i)15-s + 0.434·16-s + (0.128 − 0.991i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.267374 - 0.728646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267374 - 0.728646i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-4.41e4 + 3.41e5i)T \) |
| good | 2 | \( 1 + (24.8 - 24.8i)T - 512iT^{2} \) |
| 3 | \( 1 + (-109. - 45.3i)T + (1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (625. - 1.51e3i)T + (-1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (-4.12e3 - 9.96e3i)T + (-2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (1.95e4 - 8.11e3i)T + (1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 + 7.10e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (1.69e5 - 1.69e5i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (2.18e6 - 9.06e5i)T + (1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (5.27e5 - 1.27e6i)T + (-1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-4.36e6 - 1.80e6i)T + (1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-9.68e6 - 4.01e6i)T + (9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-2.76e6 - 6.67e6i)T + (-2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (7.19e6 + 7.19e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 - 4.00e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-1.56e7 + 1.56e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (-6.99e7 - 6.99e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (-8.20e7 - 1.98e8i)T + (-8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 + 1.65e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (2.39e7 + 9.91e6i)T + (3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-1.39e8 + 3.37e8i)T + (-4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (5.05e8 - 2.09e8i)T + (8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (4.08e8 - 4.08e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + 3.72e7iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-3.40e8 + 8.21e8i)T + (-5.37e17 - 5.37e17i)T^{2} \) |
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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
−17.91083299879020993418349789366, −15.85025135701985362840807079143, −15.15954133587775019100536682059, −14.44885536955940474290135604956, −11.78568896250781533784680722218, −9.933877684647808902683916343290, −8.625195359612702722631836361458, −7.68708234652749570233446694205, −5.88035630115720021110824692238, −2.80887818759126267036234573949,
0.49714878711412134893020998727, 1.89350821912853897601554615196, 4.10718169798081304637808204958, 7.898829508320947964365439721518, 8.508732761385911594911994016276, 10.19451503250163913639168317189, 11.45679912701607275035260277566, 12.96401155867592747374995021329, 14.25118818531794715714266831088, 16.53434381047456495027030052499
