| L(s) = 1 | + (−183. − 317. i)2-s + (−5.33e3 − 3.08e3i)3-s + (−3.42e4 + 5.93e4i)4-s + (−1.49e5 + 8.60e4i)5-s + 2.25e6i·6-s + (3.03e5 + 5.75e6i)7-s + 1.09e6·8-s + (−2.53e6 − 4.38e6i)9-s + (5.45e7 + 3.15e7i)10-s + (2.15e7 − 3.72e7i)11-s + (3.65e8 − 2.11e8i)12-s − 9.75e8i·13-s + (1.76e9 − 1.15e9i)14-s + 1.06e9·15-s + (2.04e9 + 3.54e9i)16-s + (4.97e9 + 2.87e9i)17-s + ⋯ |
| L(s) = 1 | + (−0.715 − 1.23i)2-s + (−0.813 − 0.469i)3-s + (−0.522 + 0.905i)4-s + (−0.381 + 0.220i)5-s + 1.34i·6-s + (0.0526 + 0.998i)7-s + 0.0650·8-s + (−0.0587 − 0.101i)9-s + (0.545 + 0.315i)10-s + (0.100 − 0.173i)11-s + (0.850 − 0.491i)12-s − 1.19i·13-s + (1.19 − 0.779i)14-s + 0.414·15-s + (0.476 + 0.824i)16-s + (0.713 + 0.411i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(0.500842 - 0.0870411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.500842 - 0.0870411i\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-3.03e5 - 5.75e6i)T \) |
| good | 2 | \( 1 + (183. + 317. i)T + (-3.27e4 + 5.67e4i)T^{2} \) |
| 3 | \( 1 + (5.33e3 + 3.08e3i)T + (2.15e7 + 3.72e7i)T^{2} \) |
| 5 | \( 1 + (1.49e5 - 8.60e4i)T + (7.62e10 - 1.32e11i)T^{2} \) |
| 11 | \( 1 + (-2.15e7 + 3.72e7i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 13 | \( 1 + 9.75e8iT - 6.65e17T^{2} \) |
| 17 | \( 1 + (-4.97e9 - 2.87e9i)T + (2.43e19 + 4.21e19i)T^{2} \) |
| 19 | \( 1 + (7.94e9 - 4.58e9i)T + (1.44e20 - 2.49e20i)T^{2} \) |
| 23 | \( 1 + (2.03e10 + 3.51e10i)T + (-3.06e21 + 5.31e21i)T^{2} \) |
| 29 | \( 1 - 6.70e11T + 2.50e23T^{2} \) |
| 31 | \( 1 + (-9.28e11 - 5.36e11i)T + (3.63e23 + 6.29e23i)T^{2} \) |
| 37 | \( 1 + (-2.44e12 - 4.22e12i)T + (-6.16e24 + 1.06e25i)T^{2} \) |
| 41 | \( 1 + 9.25e12iT - 6.37e25T^{2} \) |
| 43 | \( 1 + 6.77e12T + 1.36e26T^{2} \) |
| 47 | \( 1 + (-1.13e13 + 6.57e12i)T + (2.83e26 - 4.91e26i)T^{2} \) |
| 53 | \( 1 + (5.78e13 - 1.00e14i)T + (-1.93e27 - 3.35e27i)T^{2} \) |
| 59 | \( 1 + (-1.76e14 - 1.01e14i)T + (1.07e28 + 1.86e28i)T^{2} \) |
| 61 | \( 1 + (7.45e13 - 4.30e13i)T + (1.83e28 - 3.18e28i)T^{2} \) |
| 67 | \( 1 + (3.69e14 - 6.40e14i)T + (-8.24e28 - 1.42e29i)T^{2} \) |
| 71 | \( 1 - 6.18e14T + 4.16e29T^{2} \) |
| 73 | \( 1 + (1.06e15 + 6.14e14i)T + (3.25e29 + 5.63e29i)T^{2} \) |
| 79 | \( 1 + (-6.89e14 - 1.19e15i)T + (-1.15e30 + 1.99e30i)T^{2} \) |
| 83 | \( 1 - 1.94e13iT - 5.07e30T^{2} \) |
| 89 | \( 1 + (4.18e15 - 2.41e15i)T + (7.74e30 - 1.34e31i)T^{2} \) |
| 97 | \( 1 + 6.78e15iT - 6.14e31T^{2} \) |
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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
−18.38618431942934110645640997734, −17.37296162813639126515880371216, −15.21541359978353514774365603413, −12.49261173585497858463770488063, −11.74542569096223692374887287215, −10.33765248398218778211491689303, −8.447752228076871009744346420001, −5.94430358055046776783564047682, −2.96778517669952612319650993940, −1.04357148598510446082191138963,
0.40927214014025034038966058950, 4.57928504817420882536645784613, 6.44880972322049093251719925133, 7.971907970382348933881237222847, 9.870982453968396384476130750052, 11.64064714808873702611201226581, 14.17350673870097238626286626572, 15.96892158496501736917644893915, 16.67574032526119604807160724543, 17.66846249789743569980171908521
