L(s) = 1 | + (0.306 + 1.14i)2-s + i·3-s + (0.518 − 0.299i)4-s + (−0.424 + 1.58i)5-s + (−1.14 + 0.306i)6-s + (1.65 − 2.06i)7-s + (2.17 + 2.17i)8-s − 9-s − 1.94·10-s + (2.06 + 2.06i)11-s + (0.299 + 0.518i)12-s + (−2.68 + 2.40i)13-s + (2.86 + 1.26i)14-s + (−1.58 − 0.424i)15-s + (−1.22 + 2.11i)16-s + (−0.405 − 0.702i)17-s + ⋯ |
L(s) = 1 | + (0.216 + 0.808i)2-s + 0.577i·3-s + (0.259 − 0.149i)4-s + (−0.189 + 0.708i)5-s + (−0.466 + 0.125i)6-s + (0.626 − 0.779i)7-s + (0.769 + 0.769i)8-s − 0.333·9-s − 0.613·10-s + (0.621 + 0.621i)11-s + (0.0863 + 0.149i)12-s + (−0.743 + 0.668i)13-s + (0.765 + 0.338i)14-s + (−0.408 − 0.109i)15-s + (−0.305 + 0.529i)16-s + (−0.0983 − 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05749 + 1.23604i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05749 + 1.23604i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (-1.65 + 2.06i)T \) |
| 13 | \( 1 + (2.68 - 2.40i)T \) |
good | 2 | \( 1 + (-0.306 - 1.14i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (0.424 - 1.58i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.06 - 2.06i)T + 11iT^{2} \) |
| 17 | \( 1 + (0.405 + 0.702i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.56 + 4.56i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.58 + 0.917i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.42 + 4.20i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.78 + 1.01i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.95 + 1.06i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.392 + 1.46i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.14 + 1.23i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.21 + 0.325i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.12 - 1.95i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.46 - 1.73i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 - 0.516iT - 61T^{2} \) |
| 67 | \( 1 + (-9.11 + 9.11i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.21 + 8.27i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (2.64 + 9.87i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.17 + 3.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.44 - 7.44i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.18 + 8.16i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (16.5 - 4.44i)T + (84.0 - 48.5i)T^{2} \) |
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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
−11.86829169031749723831528065779, −11.10817604971474299313732252653, −10.43189928070427216567531273513, −9.344328612768592264340450528925, −7.992553108439584806099548741306, −7.06340872851658786826999961151, −6.45769330818524105475404058807, −4.90825623025624136010202474320, −4.18269416169133631281487950868, −2.26294060357442235605623796440,
1.41881288350112374265033035369, 2.69020339267525755410053745956, 4.13728066395875009515129400447, 5.45355009052743865331803168433, 6.66494823983755535924821562506, 8.001647113691786240410438331248, 8.560672800425456596420595889299, 9.922102587770723035257482898458, 11.05930898966047989353364264289, 11.79286263896319963908832717133