L(s) = 1 | + (−7 − 12.1i)5-s + (12 − 20.7i)7-s + (14 − 24.2i)11-s + (37 + 64.0i)13-s + 82·17-s + 92·19-s + (−4 − 6.92i)23-s + (−35.5 + 61.4i)25-s + (69 − 119. i)29-s + (−40 − 69.2i)31-s − 336·35-s + 30·37-s + (−141 − 244. i)41-s + (−2 + 3.46i)43-s + (−120 + 207. i)47-s + ⋯ |
L(s) = 1 | + (−0.626 − 1.08i)5-s + (0.647 − 1.12i)7-s + (0.383 − 0.664i)11-s + (0.789 + 1.36i)13-s + 1.16·17-s + 1.11·19-s + (−0.0362 − 0.0628i)23-s + (−0.284 + 0.491i)25-s + (0.441 − 0.765i)29-s + (−0.231 − 0.401i)31-s − 1.62·35-s + 0.133·37-s + (−0.537 − 0.930i)41-s + (−0.00709 + 0.0122i)43-s + (−0.372 + 0.645i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.074421594\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.074421594\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (7 + 12.1i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-12 + 20.7i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-14 + 24.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-37 - 64.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 82T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92T + 6.85e3T^{2} \) |
| 23 | \( 1 + (4 + 6.92i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-69 + 119. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (40 + 69.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 30T + 5.06e4T^{2} \) |
| 41 | \( 1 + (141 + 244. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (120 - 207. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 130T + 1.48e5T^{2} \) |
| 59 | \( 1 + (298 + 516. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-109 + 188. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-218 - 377. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 856T + 3.57e5T^{2} \) |
| 73 | \( 1 + 998T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-16 + 27.7i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-754 + 1.30e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + 246T + 7.04e5T^{2} \) |
| 97 | \( 1 + (433 - 749. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.819928736160054871634982316948, −8.961707295822698353477469615498, −8.126658211513499946455287313376, −7.49368367968520898836874632575, −6.34509132500831454100560780044, −5.14451637932726973327170862026, −4.24040063107997487079823841393, −3.55875316649750929410341632960, −1.45901289996908664180369986921, −0.71017897440555711550347109691,
1.32102376172182133613646497366, 2.86386113368538552078314028293, 3.51578422186503612533377845469, 5.05878181254525299583969326866, 5.81264931866387607073584964152, 6.96611293003743937221993075798, 7.80767048757835969635895123831, 8.468341391223881290872138516050, 9.625239550506241337803110864179, 10.48023480195193242931742421118