| L(s) = 1 | + (−0.612 + 1.27i)2-s + (0.857 + 0.514i)3-s + (−1.24 − 1.56i)4-s + (−2.17 + 0.779i)5-s + (−1.18 + 0.778i)6-s + (2.33 − 0.707i)7-s + (2.75 − 0.635i)8-s + (0.471 + 0.881i)9-s + (0.341 − 3.25i)10-s + (−0.0171 + 0.0231i)11-s + (−0.268 − 1.98i)12-s + (−3.76 + 1.78i)13-s + (−0.527 + 3.40i)14-s + (−2.27 − 0.451i)15-s + (−0.878 + 3.90i)16-s + (−5.10 + 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.433 + 0.901i)2-s + (0.495 + 0.296i)3-s + (−0.624 − 0.780i)4-s + (−0.974 + 0.348i)5-s + (−0.482 + 0.317i)6-s + (0.881 − 0.267i)7-s + (0.974 − 0.224i)8-s + (0.157 + 0.293i)9-s + (0.107 − 1.02i)10-s + (−0.00516 + 0.00697i)11-s + (−0.0775 − 0.572i)12-s + (−1.04 + 0.493i)13-s + (−0.140 + 0.910i)14-s + (−0.586 − 0.116i)15-s + (−0.219 + 0.975i)16-s + (−1.23 + 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0421237 - 0.690729i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0421237 - 0.690729i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.612 - 1.27i)T \) |
| 3 | \( 1 + (-0.857 - 0.514i)T \) |
| good | 5 | \( 1 + (2.17 - 0.779i)T + (3.86 - 3.17i)T^{2} \) |
| 7 | \( 1 + (-2.33 + 0.707i)T + (5.82 - 3.88i)T^{2} \) |
| 11 | \( 1 + (0.0171 - 0.0231i)T + (-3.19 - 10.5i)T^{2} \) |
| 13 | \( 1 + (3.76 - 1.78i)T + (8.24 - 10.0i)T^{2} \) |
| 17 | \( 1 + (5.10 - 1.01i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (0.0219 + 0.446i)T + (-18.9 + 1.86i)T^{2} \) |
| 23 | \( 1 + (-0.915 - 9.29i)T + (-22.5 + 4.48i)T^{2} \) |
| 29 | \( 1 + (0.871 - 5.87i)T + (-27.7 - 8.41i)T^{2} \) |
| 31 | \( 1 + (-4.57 - 1.89i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (4.50 + 4.97i)T + (-3.62 + 36.8i)T^{2} \) |
| 41 | \( 1 + (0.166 + 0.136i)T + (7.99 + 40.2i)T^{2} \) |
| 43 | \( 1 + (5.47 + 9.13i)T + (-20.2 + 37.9i)T^{2} \) |
| 47 | \( 1 + (0.552 + 0.369i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-0.789 - 5.32i)T + (-50.7 + 15.3i)T^{2} \) |
| 59 | \( 1 + (10.6 + 5.04i)T + (37.4 + 45.6i)T^{2} \) |
| 61 | \( 1 + (-0.0759 + 0.0190i)T + (53.7 - 28.7i)T^{2} \) |
| 67 | \( 1 + (0.0867 + 0.346i)T + (-59.0 + 31.5i)T^{2} \) |
| 71 | \( 1 + (1.28 + 0.684i)T + (39.4 + 59.0i)T^{2} \) |
| 73 | \( 1 + (-7.85 - 2.38i)T + (60.6 + 40.5i)T^{2} \) |
| 79 | \( 1 + (-8.25 - 12.3i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (2.55 - 2.81i)T + (-8.13 - 82.6i)T^{2} \) |
| 89 | \( 1 + (0.787 - 7.99i)T + (-87.2 - 17.3i)T^{2} \) |
| 97 | \( 1 + (-5.36 + 12.9i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−10.74307578215752933301068081121, −9.652948830997774279735145061057, −8.914510077730817635918822546506, −8.089382582430017237557362530345, −7.37176106665564767358501399684, −6.86091477407844350058678446059, −5.29406149585936389238799387972, −4.55332951707228468544076783837, −3.61245392370902499943754685991, −1.82014858541779171991105971973,
0.37562059392973620021258260062, 2.04397662097196010797092000680, 2.96706456810075880742200159240, 4.41438590525683954628653541483, 4.75748416321626116423800754204, 6.65836752226446543525511633508, 7.86127023258788122699625343311, 8.150864951947149214673903502587, 8.906298352862183559573585065247, 9.896138556647478988655971222940
