| L(s) = 1 | + (0.0920 − 1.05i)2-s + (2.84 + 0.500i)4-s + (−4.89 + 1.01i)5-s + (−3.71 − 2.60i)7-s + (1.88 − 7.02i)8-s + (0.612 + 5.24i)10-s + (18.1 + 6.59i)11-s + (−0.720 − 8.23i)13-s + (−3.07 + 3.66i)14-s + (3.62 + 1.32i)16-s + (−8.00 − 29.8i)17-s + (−13.8 + 8.01i)19-s + (−14.4 + 0.417i)20-s + (8.60 − 18.4i)22-s + (23.1 − 16.2i)23-s + ⋯ |
| L(s) = 1 | + (0.0460 − 0.525i)2-s + (0.710 + 0.125i)4-s + (−0.979 + 0.202i)5-s + (−0.530 − 0.371i)7-s + (0.235 − 0.877i)8-s + (0.0612 + 0.524i)10-s + (1.64 + 0.599i)11-s + (−0.0554 − 0.633i)13-s + (−0.219 + 0.262i)14-s + (0.226 + 0.0825i)16-s + (−0.470 − 1.75i)17-s + (−0.730 + 0.421i)19-s + (−0.720 + 0.0208i)20-s + (0.390 − 0.838i)22-s + (1.00 − 0.705i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0870 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0870 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.12149 - 1.22381i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12149 - 1.22381i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (4.89 - 1.01i)T \) |
| good | 2 | \( 1 + (-0.0920 + 1.05i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (3.71 + 2.60i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-18.1 - 6.59i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.720 + 8.23i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (8.00 + 29.8i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (13.8 - 8.01i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-23.1 + 16.2i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (21.0 + 25.1i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-4.78 + 27.1i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-0.000754 - 0.00281i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-27.2 - 22.9i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-10.6 - 22.8i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-26.1 - 18.3i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (24.1 + 24.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-7.25 - 19.9i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (18.4 + 104. i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-3.68 + 0.322i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (4.78 - 8.28i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (21.4 - 80.0i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-54.4 - 64.9i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (66.2 + 5.79i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (55.4 - 32.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-19.1 + 8.92i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.15551367278010042930834642192, −9.982662534959007530064054695001, −9.198940740940002814506383640126, −7.81571419994635618254063051636, −6.99590796281884927860919378597, −6.39979669317123956709675837918, −4.48839342489948051669283966542, −3.64730302736481263801799371795, −2.54193229602094303722952814913, −0.74426684569413630394981406038,
1.55474663311610910547936213987, 3.31141191363523325526117244400, 4.32476832824267738446223843011, 5.83297717946794980418984232576, 6.62577798417846583991405731979, 7.31879162664807180038141756777, 8.690098057621832292868452976467, 8.985127709665103886778119534480, 10.70129546475669624499004153095, 11.25805691530758323806388239547
