L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.692·5-s + (−0.499 + 0.866i)6-s + (0.178 − 0.309i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.346 − 0.599i)10-s + (1.46 + 2.54i)11-s + 0.999·12-s − 0.356·14-s + (−0.346 − 0.599i)15-s + (−0.5 − 0.866i)16-s + (−3.35 + 5.81i)17-s + 0.999·18-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.309·5-s + (−0.204 + 0.353i)6-s + (0.0674 − 0.116i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.109 − 0.189i)10-s + (0.443 + 0.767i)11-s + 0.288·12-s − 0.0953·14-s + (−0.0893 − 0.154i)15-s + (−0.125 − 0.216i)16-s + (−0.814 + 1.41i)17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7921169620\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7921169620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.692T + 5T^{2} \) |
| 7 | \( 1 + (-0.178 + 0.309i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.46 - 2.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (3.35 - 5.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.60 - 6.24i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.19 + 2.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.91 + 6.78i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + (-5.04 - 8.74i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.44 - 4.23i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.29 - 5.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 + (-0.821 + 1.42i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 + 5.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.76 - 11.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.40 + 5.89i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + (-0.198 - 0.343i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.208 - 0.361i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.02223700850639568736204295007, −9.511060222882615275313327435366, −8.157084825513661323158219601008, −7.972878399749771210342778106405, −6.47977112717427163651964414862, −6.11455031680885079715621713128, −4.59232487405920705002323716809, −3.84283348507595269752796951266, −2.25588302107791521986580938069, −1.52152808306359599719938655724,
0.41978786349400754228209308620, 2.28191041168108284732690119978, 3.72018216350965597171951488138, 4.82790998371192219141999923997, 5.55327332760819123227796474383, 6.50328984777779055481977328938, 7.17511305749338941203468955384, 8.328869402437628363300986673132, 9.277330309580613008299257837999, 9.392263161449641825605908034639