| L(s) = 1 | + (−8.75 − 15.1i)2-s + (−30.3 − 35.6i)3-s + (−89.2 + 154. i)4-s + (242. − 419. i)5-s + (−274. + 771. i)6-s + (−171.5 − 297. i)7-s + 885.·8-s + (−349. + 2.15e3i)9-s − 8.48e3·10-s + (779. + 1.34e3i)11-s + (8.21e3 − 1.50e3i)12-s + (−6.43e3 + 1.11e4i)13-s + (−3.00e3 + 5.20e3i)14-s + (−2.22e4 + 4.09e3i)15-s + (3.67e3 + 6.36e3i)16-s + 1.35e4·17-s + ⋯ |
| L(s) = 1 | + (−0.773 − 1.34i)2-s + (−0.648 − 0.761i)3-s + (−0.697 + 1.20i)4-s + (0.867 − 1.50i)5-s + (−0.519 + 1.45i)6-s + (−0.188 − 0.327i)7-s + 0.611·8-s + (−0.159 + 0.987i)9-s − 2.68·10-s + (0.176 + 0.305i)11-s + (1.37 − 0.251i)12-s + (−0.812 + 1.40i)13-s + (−0.292 + 0.506i)14-s + (−1.70 + 0.313i)15-s + (0.224 + 0.388i)16-s + 0.668·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0141i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.110277 + 0.000778227i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.110277 + 0.000778227i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (30.3 + 35.6i)T \) |
| 7 | \( 1 + (171.5 + 297. i)T \) |
| good | 2 | \( 1 + (8.75 + 15.1i)T + (-64 + 110. i)T^{2} \) |
| 5 | \( 1 + (-242. + 419. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-779. - 1.34e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (6.43e3 - 1.11e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.35e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.81e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (4.20e4 - 7.28e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (5.28e4 + 9.15e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-6.86e4 + 1.18e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.44e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (9.97e4 - 1.72e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-4.05e5 - 7.01e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-2.97e5 - 5.14e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.84e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-8.24e5 + 1.42e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (9.91e5 + 1.71e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.68e5 - 4.65e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.85e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.55e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (9.15e5 + 1.58e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.86e6 - 4.96e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 3.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.02e5 + 3.50e5i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.95252378541249433988149279015, −12.24317742378459027831216014848, −11.43743313292330210766575966771, −9.900167296259189186433893871276, −9.300520833294399554298761509231, −7.891073990937244362612861919993, −6.06958730060397181918010321830, −4.51927427168953276214589666188, −2.02653537765451099242302725465, −1.29212961748470927660271159140,
0.05902122115011443176972337194, 3.01192611828824422347616387509, 5.46089958404344092160101573983, 6.19624477322929972898990540493, 7.22541105820751837540663464882, 8.820238803516563546455561031098, 10.18260672316855929036644622074, 10.44409170974104863833654610650, 12.26803802092806251014819239202, 14.25219248641726789751685786250
