L(s) = 1 | − 1.90·2-s + (−0.214 − 0.371i)3-s + 1.63·4-s + (0.736 + 1.27i)5-s + (0.408 + 0.707i)6-s + (1.58 − 2.11i)7-s + 0.702·8-s + (1.40 − 2.43i)9-s + (−1.40 − 2.43i)10-s + (2.19 + 3.80i)11-s + (−0.349 − 0.605i)12-s + (2.69 − 2.39i)13-s + (−3.01 + 4.03i)14-s + (0.315 − 0.546i)15-s − 4.60·16-s − 1.20·17-s + ⋯ |
L(s) = 1 | − 1.34·2-s + (−0.123 − 0.214i)3-s + 0.815·4-s + (0.329 + 0.570i)5-s + (0.166 + 0.288i)6-s + (0.598 − 0.801i)7-s + 0.248·8-s + (0.469 − 0.813i)9-s + (−0.443 − 0.768i)10-s + (0.662 + 1.14i)11-s + (−0.100 − 0.174i)12-s + (0.748 − 0.663i)13-s + (−0.806 + 1.07i)14-s + (0.0814 − 0.141i)15-s − 1.15·16-s − 0.291·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.576458 - 0.0717422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.576458 - 0.0717422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.58 + 2.11i)T \) |
| 13 | \( 1 + (-2.69 + 2.39i)T \) |
good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 3 | \( 1 + (0.214 + 0.371i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.736 - 1.27i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 19 | \( 1 + (1.62 - 2.80i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 + (0.0837 - 0.145i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.62 - 4.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.05T + 37T^{2} \) |
| 41 | \( 1 + (2.58 - 4.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0113 + 0.0197i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.84 + 10.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0708 + 0.122i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 5.34T + 59T^{2} \) |
| 61 | \( 1 + (5.77 - 9.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.98 - 8.63i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.62 - 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 6.55T + 89T^{2} \) |
| 97 | \( 1 + (1.74 + 3.02i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.19312166818781083484072408101, −12.92277519425206551263718891857, −11.58214372436119730036680721030, −10.39694015094341403308773024683, −9.864657149160381039803818026450, −8.512679179293615699492156229588, −7.36875551612933266975547147177, −6.47607079543751022548417620216, −4.18697088988592255677401499061, −1.51908533655827400352476466536,
1.72563343605294342870054241683, 4.57519299751439993358271528029, 6.12268935248963430456661375642, 7.84636563311050374250092532007, 8.799277999682287982094250276349, 9.407300660985387741089847723931, 10.87888075105104611806449727922, 11.46430224234113307168011959196, 13.13320571208889022710442854228, 14.06847494399047882914994779317