| L(s) = 1 | + (−9.35 − 16.1i)2-s + (31.2 + 54.1i)3-s + (−110. + 192. i)4-s − 263.·5-s + (584. − 1.01e3i)6-s + (171.5 − 297. i)7-s + 1.75e3·8-s + (−859. + 1.48e3i)9-s + (2.46e3 + 4.27e3i)10-s + (−1.94e3 − 3.37e3i)11-s − 1.38e4·12-s + (−6.64e3 − 4.30e3i)13-s − 6.41e3·14-s + (−8.24e3 − 1.42e4i)15-s + (−2.21e3 − 3.84e3i)16-s + (9.63e3 − 1.66e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.826 − 1.43i)2-s + (0.668 + 1.15i)3-s + (−0.866 + 1.50i)4-s − 0.943·5-s + (1.10 − 1.91i)6-s + (0.188 − 0.327i)7-s + 1.21·8-s + (−0.392 + 0.680i)9-s + (0.780 + 1.35i)10-s + (−0.441 − 0.764i)11-s − 2.31·12-s + (−0.839 − 0.544i)13-s − 0.624·14-s + (−0.630 − 1.09i)15-s + (−0.135 − 0.234i)16-s + (0.475 − 0.824i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.953 + 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.999729 - 0.154684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999729 - 0.154684i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 + 297. i)T \) |
| 13 | \( 1 + (6.64e3 + 4.30e3i)T \) |
| good | 2 | \( 1 + (9.35 + 16.1i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (-31.2 - 54.1i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 263.T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.94e3 + 3.37e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (-9.63e3 + 1.66e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (6.79e3 - 1.17e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.75e4 - 8.23e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-7.70e4 - 1.33e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.94e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.32e5 - 2.30e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-6.34e3 - 1.09e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.44e5 - 2.49e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 3.36e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.05e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.09e5 + 1.89e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.23e5 - 7.33e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.10e6 - 1.90e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.77e6 + 4.80e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + 8.85e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.82e6 - 8.35e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-6.78e6 + 1.17e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−12.13511805006655099469609945171, −11.29563770429813133105029691057, −10.33585783641498124417307816820, −9.630563377251553870619710946237, −8.529926061506908503371394347700, −7.69198400072658191890755691943, −4.85057945035174349117278385677, −3.53627530093405307126759877981, −2.90452412759290902423360688514, −0.824400918364549295607405300362,
0.58661557344449738009012149415, 2.35997043756366032476490523323, 4.69706788620792330088137396134, 6.43983074063423215464051886022, 7.32524163819767646563638909378, 8.012268389484840359291661395651, 8.731750852467946305909184740890, 10.10991644199473460453696294567, 11.93190115691344622953711212225, 12.82032790492668165261015974755
