| L(s) = 1 | + (−10.7 − 10.7i)2-s − 23.0i·4-s + (489. − 388. i)5-s + (2.35e3 + 2.35e3i)7-s + (−3.01e3 + 3.01e3i)8-s + (−9.47e3 − 1.09e3i)10-s + 1.19e4·11-s + (−3.71e4 + 3.71e4i)13-s − 5.08e4i·14-s + 5.91e4·16-s + (7.70e4 + 7.70e4i)17-s + 1.72e5i·19-s + (−8.93e3 − 1.12e4i)20-s + (−1.28e5 − 1.28e5i)22-s + (3.03e5 − 3.03e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.674 − 0.674i)2-s − 0.0899i·4-s + (0.783 − 0.621i)5-s + (0.980 + 0.980i)7-s + (−0.735 + 0.735i)8-s + (−0.947 − 0.109i)10-s + 0.813·11-s + (−1.30 + 1.30i)13-s − 1.32i·14-s + 0.902·16-s + (0.922 + 0.922i)17-s + 1.32i·19-s + (−0.0558 − 0.0704i)20-s + (−0.549 − 0.549i)22-s + (1.08 − 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.56939 - 0.274806i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56939 - 0.274806i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-489. + 388. i)T \) |
| good | 2 | \( 1 + (10.7 + 10.7i)T + 256iT^{2} \) |
| 7 | \( 1 + (-2.35e3 - 2.35e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 - 1.19e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (3.71e4 - 3.71e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (-7.70e4 - 7.70e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 - 1.72e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-3.03e5 + 3.03e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + 3.61e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 4.32e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-6.98e5 - 6.98e5i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + 2.69e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (-3.37e5 + 3.37e5i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-1.97e6 - 1.97e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (3.56e6 - 3.56e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 + 9.30e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.35e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-2.69e5 - 2.69e5i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 4.06e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (7.18e5 - 7.18e5i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 2.64e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-5.69e7 + 5.69e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 1.47e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-5.88e7 - 5.88e7i)T + 7.83e15iT^{2} \) |
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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
−14.38763979280341763025042649066, −12.33235416811264055411746470191, −11.74116769719700668923389431869, −10.20108752218966830699518816525, −9.252111199585692139313736923641, −8.356763102356558748535070549118, −6.09945461865659739107917174564, −4.84747875607000516506130090867, −2.19234807067568214271218608619, −1.34160417781755505221814825541,
0.861001172932950504661313928715, 3.05822542544720531564414892698, 5.17004229697935474319022716276, 7.00793403183732294256940605802, 7.59553327078832910919761972563, 9.228059809273938604587007052457, 10.26245740122594398771933845043, 11.62938010300015106890204079875, 13.19989102455267238521107055699, 14.39897539346215907208134604647
