| L(s) = 1 | + (1.14 + 0.825i)2-s + (−0.0980 + 0.995i)3-s + (0.635 + 1.89i)4-s + (1.92 − 1.03i)5-s + (−0.934 + 1.06i)6-s + (0.264 + 1.32i)7-s + (−0.836 + 2.70i)8-s + (−0.980 − 0.195i)9-s + (3.06 + 0.409i)10-s + (−1.22 + 1.49i)11-s + (−1.94 + 0.446i)12-s + (1.87 − 3.50i)13-s + (−0.793 + 1.74i)14-s + (0.837 + 2.02i)15-s + (−3.19 + 2.41i)16-s + (0.375 − 0.907i)17-s + ⋯ |
| L(s) = 1 | + (0.811 + 0.584i)2-s + (−0.0565 + 0.574i)3-s + (0.317 + 0.948i)4-s + (0.862 − 0.461i)5-s + (−0.381 + 0.433i)6-s + (0.0998 + 0.502i)7-s + (−0.295 + 0.955i)8-s + (−0.326 − 0.0650i)9-s + (0.969 + 0.129i)10-s + (−0.370 + 0.451i)11-s + (−0.562 + 0.128i)12-s + (0.519 − 0.971i)13-s + (−0.212 + 0.465i)14-s + (0.216 + 0.521i)15-s + (−0.797 + 0.602i)16-s + (0.0911 − 0.220i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00313 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00313 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61770 + 1.62277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61770 + 1.62277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.14 - 0.825i)T \) |
| 3 | \( 1 + (0.0980 - 0.995i)T \) |
| good | 5 | \( 1 + (-1.92 + 1.03i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.264 - 1.32i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (1.22 - 1.49i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.87 + 3.50i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.375 + 0.907i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.235 + 0.777i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (0.306 - 0.204i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.252 + 0.207i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-2.39 + 2.39i)T - 31iT^{2} \) |
| 37 | \( 1 + (1.23 + 0.373i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (6.10 + 9.13i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.421 - 4.28i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (4.16 + 1.72i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.96 - 1.61i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (5.62 + 10.5i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-9.21 - 0.907i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 1.33i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-3.68 + 0.733i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.22 + 6.15i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (2.47 - 1.02i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-13.7 + 4.17i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (8.67 + 5.79i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (1.58 - 1.58i)T - 97iT^{2} \) |
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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
−11.74366418916979858190884754218, −10.68752058317717327515132100689, −9.679298601021652031058102406548, −8.723708937834378371096556955480, −7.85960299867251168312784393561, −6.51832315739295935157648793106, −5.48620123923582800365680180257, −5.05943117056858645077465590751, −3.65523596472831316548333870675, −2.33089472771983698931839186621,
1.45731514348391421460372180775, 2.65231513373414804955688468817, 3.95716607304202155366525981286, 5.30234624319973075287390978327, 6.29744010912382059021964511293, 6.88898614077173347888789435813, 8.331713293839701836970138207134, 9.607267270691165337436529303868, 10.43592656276437687644571024557, 11.17756593205684982369093544703
