| L(s) = 1 | + (0.468 − 0.485i)2-s + (1.59 − 2.61i)3-s + (0.129 + 3.54i)4-s + (1.20 + 1.68i)5-s + (−0.523 − 2.00i)6-s + (8.95 − 4.60i)7-s + (3.78 + 3.39i)8-s + (−0.190 − 0.369i)9-s + (1.37 + 0.203i)10-s + (−6.35 + 4.20i)11-s + (9.48 + 5.32i)12-s + (4.93 − 4.75i)13-s + (1.95 − 6.49i)14-s + (6.32 − 0.463i)15-s + (−10.7 + 0.784i)16-s + (9.97 + 15.0i)17-s + ⋯ |
| L(s) = 1 | + (0.234 − 0.242i)2-s + (0.532 − 0.873i)3-s + (0.0323 + 0.885i)4-s + (0.240 + 0.336i)5-s + (−0.0872 − 0.333i)6-s + (1.27 − 0.657i)7-s + (0.473 + 0.424i)8-s + (−0.0211 − 0.0411i)9-s + (0.137 + 0.0203i)10-s + (−0.577 + 0.382i)11-s + (0.790 + 0.443i)12-s + (0.379 − 0.366i)13-s + (0.139 − 0.464i)14-s + (0.421 − 0.0308i)15-s + (−0.670 + 0.0490i)16-s + (0.586 + 0.886i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.82300 - 0.229247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.82300 - 0.229247i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 431 | \( 1 + (134. + 409. i)T \) |
| good | 2 | \( 1 + (-0.468 + 0.485i)T + (-0.146 - 3.99i)T^{2} \) |
| 3 | \( 1 + (-1.59 + 2.61i)T + (-4.11 - 8.00i)T^{2} \) |
| 5 | \( 1 + (-1.20 - 1.68i)T + (-8.07 + 23.6i)T^{2} \) |
| 7 | \( 1 + (-8.95 + 4.60i)T + (28.5 - 39.8i)T^{2} \) |
| 11 | \( 1 + (6.35 - 4.20i)T + (47.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (-4.93 + 4.75i)T + (6.17 - 168. i)T^{2} \) |
| 17 | \( 1 + (-9.97 - 15.0i)T + (-113. + 265. i)T^{2} \) |
| 19 | \( 1 + (14.2 - 19.8i)T + (-116. - 341. i)T^{2} \) |
| 23 | \( 1 + (-9.67 + 11.6i)T + (-96.0 - 520. i)T^{2} \) |
| 29 | \( 1 + (0.100 - 2.74i)T + (-838. - 61.3i)T^{2} \) |
| 31 | \( 1 + (-5.22 - 1.36i)T + (838. + 470. i)T^{2} \) |
| 37 | \( 1 + (-4.46 + 14.8i)T + (-1.14e3 - 755. i)T^{2} \) |
| 41 | \( 1 + (-32.6 + 2.38i)T + (1.66e3 - 244. i)T^{2} \) |
| 43 | \( 1 + (7.83 + 53.2i)T + (-1.77e3 + 532. i)T^{2} \) |
| 47 | \( 1 + (-8.23 - 21.5i)T + (-1.64e3 + 1.47e3i)T^{2} \) |
| 53 | \( 1 + (47.7 + 18.2i)T + (2.09e3 + 1.87e3i)T^{2} \) |
| 59 | \( 1 + (-20.3 + 21.1i)T + (-127. - 3.47e3i)T^{2} \) |
| 61 | \( 1 + (60.7 + 28.4i)T + (2.38e3 + 2.86e3i)T^{2} \) |
| 67 | \( 1 + (-4.09 + 13.6i)T + (-3.74e3 - 2.47e3i)T^{2} \) |
| 71 | \( 1 + (23.9 + 62.6i)T + (-3.75e3 + 3.36e3i)T^{2} \) |
| 73 | \( 1 + (3.79 + 17.0i)T + (-4.82e3 + 2.26e3i)T^{2} \) |
| 79 | \( 1 + (113. + 12.5i)T + (6.09e3 + 1.35e3i)T^{2} \) |
| 83 | \( 1 + (8.34 + 5.09i)T + (3.14e3 + 6.12e3i)T^{2} \) |
| 89 | \( 1 + (-42.4 - 54.9i)T + (-2.00e3 + 7.66e3i)T^{2} \) |
| 97 | \( 1 + (-17.7 - 161. i)T + (-9.18e3 + 2.04e3i)T^{2} \) |
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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
−10.72900049844882353721529739571, −10.52047648429413114470863180068, −8.655368364097344414707468200866, −7.902165340098530654130890873760, −7.64210210717608029592189704966, −6.43605110345122437718080741280, −4.90798676346583394803043143658, −3.86445091604725586348798161434, −2.51306077297812044347728201857, −1.57860583774240031244119104098,
1.29529483612786813411561403737, 2.79761797217816188597159997668, 4.46804124654510730770525369076, 5.02756596342150210639032490781, 5.92519063750801597950612984052, 7.26657261845087847133644149841, 8.537716035022160365458160245904, 9.157118234635019351429834242493, 9.927067122220901717454135921620, 10.99564979566923912473378329689
