L(s) = 1 | + (−107. + 232. i)2-s + 693.·3-s + (−4.23e4 − 5.00e4i)4-s + 3.33e5i·5-s + (−7.46e4 + 1.60e5i)6-s − 1.01e7i·7-s + (1.61e7 − 4.43e6i)8-s − 4.25e7·9-s + (−7.73e7 − 3.58e7i)10-s + 3.53e8·11-s + (−2.93e7 − 3.46e7i)12-s + 4.13e8i·13-s + (2.35e9 + 1.09e9i)14-s + 2.30e8i·15-s + (−7.12e8 + 4.23e9i)16-s + 7.22e9·17-s + ⋯ |
L(s) = 1 | + (−0.420 + 0.907i)2-s + 0.105·3-s + (−0.645 − 0.763i)4-s + 0.852i·5-s + (−0.0444 + 0.0958i)6-s − 1.75i·7-s + (0.964 − 0.264i)8-s − 0.988·9-s + (−0.773 − 0.358i)10-s + 1.65·11-s + (−0.0682 − 0.0806i)12-s + 0.506i·13-s + (1.59 + 0.739i)14-s + 0.0900i·15-s + (−0.165 + 0.986i)16-s + 1.03·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(1.37501 + 0.185198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37501 + 0.185198i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (107. - 232. i)T \) |
good | 3 | \( 1 - 693.T + 4.30e7T^{2} \) |
| 5 | \( 1 - 3.33e5iT - 1.52e11T^{2} \) |
| 7 | \( 1 + 1.01e7iT - 3.32e13T^{2} \) |
| 11 | \( 1 - 3.53e8T + 4.59e16T^{2} \) |
| 13 | \( 1 - 4.13e8iT - 6.65e17T^{2} \) |
| 17 | \( 1 - 7.22e9T + 4.86e19T^{2} \) |
| 19 | \( 1 - 1.17e10T + 2.88e20T^{2} \) |
| 23 | \( 1 + 3.06e10iT - 6.13e21T^{2} \) |
| 29 | \( 1 + 4.93e11iT - 2.50e23T^{2} \) |
| 31 | \( 1 + 8.90e11iT - 7.27e23T^{2} \) |
| 37 | \( 1 + 2.63e12iT - 1.23e25T^{2} \) |
| 41 | \( 1 - 1.13e13T + 6.37e25T^{2} \) |
| 43 | \( 1 + 6.39e11T + 1.36e26T^{2} \) |
| 47 | \( 1 - 9.74e12iT - 5.66e26T^{2} \) |
| 53 | \( 1 - 3.29e13iT - 3.87e27T^{2} \) |
| 59 | \( 1 + 7.37e13T + 2.15e28T^{2} \) |
| 61 | \( 1 + 3.01e14iT - 3.67e28T^{2} \) |
| 67 | \( 1 - 6.34e14T + 1.64e29T^{2} \) |
| 71 | \( 1 + 3.99e12iT - 4.16e29T^{2} \) |
| 73 | \( 1 + 7.97e14T + 6.50e29T^{2} \) |
| 79 | \( 1 - 9.10e14iT - 2.30e30T^{2} \) |
| 83 | \( 1 - 7.04e14T + 5.07e30T^{2} \) |
| 89 | \( 1 + 1.16e15T + 1.54e31T^{2} \) |
| 97 | \( 1 + 3.41e15T + 6.14e31T^{2} \) |
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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
−17.34585679796634302791140883461, −16.64314449716451430334209963761, −14.42944033042282287852281177915, −14.06342473322024135567999782708, −11.11086125484475536996415172605, −9.577676994220679805641426636965, −7.59373667847457654408699269868, −6.37578247815655809153238164908, −3.92448794065837812990122188108, −0.842620766893523494102672025170,
1.26839501227598652429847617434, 3.09065411220274613576488858269, 5.38653380030244506637117964229, 8.536481487129183713332045062116, 9.294775551575173411711052339752, 11.69104088322318012776738774883, 12.42528428727354869938261018541, 14.44171760384715399810190727008, 16.50026610747508433806853379912, 17.81774395866576792243438194006