| L(s) = 1 | + (0.537 − 0.0755i)2-s + (−1.63 + 0.470i)4-s + (−3.32 + 1.34i)5-s + (0.885 + 0.597i)7-s + (−1.83 + 0.817i)8-s + (−1.68 + 0.974i)10-s + (−1.29 − 3.05i)11-s + (1.23 + 2.32i)13-s + (0.520 + 0.253i)14-s + (1.96 − 1.22i)16-s + (1.71 − 0.365i)17-s + (−0.0216 − 0.0486i)19-s + (4.82 − 3.76i)20-s + (−0.924 − 1.54i)22-s + (2.86 − 7.88i)23-s + ⋯ |
| L(s) = 1 | + (0.379 − 0.0534i)2-s + (−0.819 + 0.235i)4-s + (−1.48 + 0.601i)5-s + (0.334 + 0.225i)7-s + (−0.649 + 0.289i)8-s + (−0.533 + 0.308i)10-s + (−0.389 − 0.921i)11-s + (0.342 + 0.644i)13-s + (0.139 + 0.0678i)14-s + (0.491 − 0.307i)16-s + (0.416 − 0.0885i)17-s + (−0.00497 − 0.0111i)19-s + (1.07 − 0.842i)20-s + (−0.197 − 0.329i)22-s + (0.598 − 1.64i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.546658 - 0.426402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.546658 - 0.426402i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.29 + 3.05i)T \) |
| good | 2 | \( 1 + (-0.537 + 0.0755i)T + (1.92 - 0.551i)T^{2} \) |
| 5 | \( 1 + (3.32 - 1.34i)T + (3.59 - 3.47i)T^{2} \) |
| 7 | \( 1 + (-0.885 - 0.597i)T + (2.62 + 6.49i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 2.32i)T + (-7.26 + 10.7i)T^{2} \) |
| 17 | \( 1 + (-1.71 + 0.365i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0216 + 0.0486i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.86 + 7.88i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (3.86 + 7.91i)T + (-17.8 + 22.8i)T^{2} \) |
| 31 | \( 1 + (0.0700 - 2.00i)T + (-30.9 - 2.16i)T^{2} \) |
| 37 | \( 1 + (3.34 + 1.48i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (4.76 - 9.76i)T + (-25.2 - 32.3i)T^{2} \) |
| 43 | \( 1 + (-6.03 + 7.19i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.158 + 0.552i)T + (-39.8 - 24.9i)T^{2} \) |
| 53 | \( 1 + (0.862 - 0.280i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-7.68 - 1.91i)T + (52.0 + 27.6i)T^{2} \) |
| 61 | \( 1 + (-10.3 + 0.362i)T + (60.8 - 4.25i)T^{2} \) |
| 67 | \( 1 + (0.424 + 2.40i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.524 + 2.46i)T + (-64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (9.10 - 0.956i)T + (71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (0.309 + 2.20i)T + (-75.9 + 21.7i)T^{2} \) |
| 83 | \( 1 + (8.66 + 4.60i)T + (46.4 + 68.8i)T^{2} \) |
| 89 | \( 1 + (-8.64 - 4.99i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.43 - 6.02i)T + (-69.7 - 67.3i)T^{2} \) |
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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
−10.03035870064631579883414794444, −8.732907079619733071292628849221, −8.382457285074695139150387759109, −7.55326827989263537359891783502, −6.50850135696679657005091918237, −5.37634446778570036704779847395, −4.36474963633499503136249225586, −3.68566573454550634701616867899, −2.76110314767928842467885471773, −0.35943529039724829466413600319,
1.17350164674045800009935707843, 3.36993210893997067179700265316, 4.02617833854592050744283542900, 4.98412081956595629240284454774, 5.51243328977371696876810477228, 7.16038542983462950103330833491, 7.76447964639682859545434622197, 8.569034734644541357991836622112, 9.336701653531120231155003234496, 10.31125265417131513386260918594
