| L(s) = 1 | + (0.886 − 3.30i)2-s + (−0.0164 − 0.0285i)3-s + (−6.68 − 3.86i)4-s + (−1.58 − 1.58i)5-s + (−0.109 + 0.0292i)6-s + (0.185 + 0.692i)7-s + (−9.01 + 9.01i)8-s + (4.49 − 7.79i)9-s + (−6.63 + 3.82i)10-s + (15.5 + 4.17i)11-s + 0.254i·12-s + (−2.16 + 12.8i)13-s + 2.45·14-s + (−0.0190 + 0.0712i)15-s + (6.38 + 11.0i)16-s + (−1.49 − 0.860i)17-s + ⋯ |
| L(s) = 1 | + (0.443 − 1.65i)2-s + (−0.00549 − 0.00951i)3-s + (−1.67 − 0.965i)4-s + (−0.316 − 0.316i)5-s + (−0.0181 + 0.00487i)6-s + (0.0265 + 0.0989i)7-s + (−1.12 + 1.12i)8-s + (0.499 − 0.865i)9-s + (−0.663 + 0.382i)10-s + (1.41 + 0.379i)11-s + 0.0212i·12-s + (−0.166 + 0.986i)13-s + 0.175·14-s + (−0.00127 + 0.00474i)15-s + (0.399 + 0.691i)16-s + (−0.0876 − 0.0506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.444354 - 1.30371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.444354 - 1.30371i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.58 + 1.58i)T \) |
| 13 | \( 1 + (2.16 - 12.8i)T \) |
| good | 2 | \( 1 + (-0.886 + 3.30i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (0.0164 + 0.0285i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (-0.185 - 0.692i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-15.5 - 4.17i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (1.49 + 0.860i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.58 + 1.76i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-1.16 + 0.674i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-23.7 - 41.1i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (21.6 + 21.6i)T + 961iT^{2} \) |
| 37 | \( 1 + (42.2 + 11.3i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (14.6 - 54.8i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-22.7 - 13.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (57.0 - 57.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 31.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (24.3 + 90.7i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (21.4 - 37.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-25.5 + 95.4i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-50.7 + 13.5i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (52.6 - 52.6i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 41.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-8.45 - 8.45i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-91.6 - 24.5i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (141. - 37.8i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−13.97565666740252341552280246306, −12.61570423566964674868038105394, −12.04005855477189608459503514399, −11.18049639939025886011620482559, −9.671410365966896280161537509871, −9.043985964266311743586727905637, −6.77729763460227217506661004764, −4.64384584070057176481240203009, −3.59599744950960521972315809720, −1.45131721909569642300070207169,
3.93356965564017059167105120703, 5.36249270645372987244693748905, 6.70575209608892825550082531352, 7.66948948661750653089551121356, 8.769825392599784721274830770308, 10.42937965264582162240097779595, 12.03526957535246493207370697509, 13.42365935841921947724058371382, 14.16040626805972397476174483902, 15.18542147625910968594739181323
