L(s) = 1 | + 4.24i·2-s + 238·4-s + 988. i·5-s + 1.65e3·7-s + 2.09e3i·8-s − 4.19e3·10-s − 2.13e4i·11-s − 2.62e4·13-s + 7.00e3i·14-s + 5.20e4·16-s + 1.51e4i·17-s + 4.66e4·19-s + 2.35e5i·20-s + 9.06e4·22-s − 3.28e5i·23-s + ⋯ |
L(s) = 1 | + 0.265i·2-s + 0.929·4-s + 1.58i·5-s + 0.688·7-s + 0.511i·8-s − 0.419·10-s − 1.45i·11-s − 0.919·13-s + 0.182i·14-s + 0.794·16-s + 0.181i·17-s + 0.357·19-s + 1.47i·20-s + 0.386·22-s − 1.17i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.53969 + 0.797003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53969 + 0.797003i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 - 4.24iT - 256T^{2} \) |
| 5 | \( 1 - 988. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.65e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.13e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.62e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.51e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 4.66e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.28e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 6.14e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.96e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.81e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 7.06e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.21e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.63e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 5.18e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.17e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.74e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.43e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.56e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 8.90e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.27e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 8.50e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 5.89e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 2.44e7T + 7.83e15T^{2} \) |
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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
−19.54184962159593855834933396899, −18.33929053281741577534652709849, −16.71617980069690576476837497665, −15.10883559580278349229824119286, −14.20501142008382155188432796360, −11.56685349254782266174603068421, −10.60025129642198135097972432621, −7.74053703127171466354489329985, −6.22974748587660468506079094773, −2.73750338537879341949836936406,
1.63018596306263491465773435880, 4.92784206874732720909095783752, 7.59476956713441111806571926180, 9.644242888902663933272328095892, 11.74151011440094799573884440791, 12.75865203634560429199435592488, 15.01144269951227098735604480782, 16.41174529581820502358707449725, 17.59442194777383088836091679751, 19.88128048563592376656692651939