L(s) = 1 | + (11.8 + 11.8i)2-s + (11.8 − 11.8i)3-s + 26i·4-s + 282·6-s + (2.74e3 + 2.74e3i)7-s + (2.73e3 − 2.73e3i)8-s + 6.27e3i·9-s + 1.21e4·11-s + (308. + 308. i)12-s + (−2.42e3 + 2.42e3i)13-s + 6.51e4i·14-s + 7.15e4·16-s + (−7.69e4 − 7.69e4i)17-s + (−7.45e4 + 7.45e4i)18-s + 1.68e5i·19-s + ⋯ |
L(s) = 1 | + (0.742 + 0.742i)2-s + (0.146 − 0.146i)3-s + 0.101i·4-s + 0.217·6-s + (1.14 + 1.14i)7-s + (0.666 − 0.666i)8-s + 0.957i·9-s + 0.828·11-s + (0.0148 + 0.0148i)12-s + (−0.0848 + 0.0848i)13-s + 1.69i·14-s + 1.09·16-s + (−0.921 − 0.921i)17-s + (−0.710 + 0.710i)18-s + 1.29i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.64012 + 1.47196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.64012 + 1.47196i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-11.8 - 11.8i)T + 256iT^{2} \) |
| 3 | \( 1 + (-11.8 + 11.8i)T - 6.56e3iT^{2} \) |
| 7 | \( 1 + (-2.74e3 - 2.74e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 - 1.21e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (2.42e3 - 2.42e3i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (7.69e4 + 7.69e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 - 1.68e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-2.21e5 + 2.21e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + 6.66e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.04e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + (2.07e6 + 2.07e6i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + 1.32e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (-2.78e6 + 2.78e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-3.63e6 - 3.63e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (3.14e6 - 3.14e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 - 6.49e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.43e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (1.14e7 + 1.14e7i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 2.30e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.74e7 + 1.74e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 2.76e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (1.15e7 - 1.15e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 2.61e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (8.09e7 + 8.09e7i)T + 7.83e15iT^{2} \) |
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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
−15.65088868580879044074156668235, −14.59346555028032018232909106134, −13.86366456730445677262521921083, −12.31394749832771367883096243616, −10.89270854844029005366075275826, −8.909358550712252079513390698647, −7.42269233055240161981700020501, −5.74908006895450568341471261491, −4.58818484252313081043069292160, −1.90749713065090888877954839981,
1.42436558255833054154923093605, 3.57094785846168390746734024769, 4.68859398590647797210003383307, 7.10231960563365664466626627171, 8.810606409920519879239672108156, 10.75923517528193821773334966735, 11.59787917445806493331916721336, 13.00374424282089135690121423749, 14.10427632742838676061948696480, 15.08357287324134628993829348232