L(s) = 1 | + (−2.35 − 0.631i)2-s + (1.72 − 0.102i)3-s + (3.42 + 1.97i)4-s + (0.0540 + 2.23i)5-s + (−4.13 − 0.849i)6-s + (−1.91 + 1.82i)7-s + (−3.36 − 3.36i)8-s + (2.97 − 0.355i)9-s + (1.28 − 5.30i)10-s + (3.08 + 1.77i)11-s + (6.11 + 3.06i)12-s + (1.28 − 1.28i)13-s + (5.67 − 3.08i)14-s + (0.323 + 3.85i)15-s + (1.85 + 3.21i)16-s + (−0.792 − 2.95i)17-s + ⋯ |
L(s) = 1 | + (−1.66 − 0.446i)2-s + (0.998 − 0.0593i)3-s + (1.71 + 0.987i)4-s + (0.0241 + 0.999i)5-s + (−1.68 − 0.346i)6-s + (−0.725 + 0.688i)7-s + (−1.19 − 1.19i)8-s + (0.992 − 0.118i)9-s + (0.406 − 1.67i)10-s + (0.928 + 0.536i)11-s + (1.76 + 0.884i)12-s + (0.356 − 0.356i)13-s + (1.51 − 0.823i)14-s + (0.0834 + 0.996i)15-s + (0.463 + 0.803i)16-s + (−0.192 − 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.654808 + 0.0942559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.654808 + 0.0942559i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 + 0.102i)T \) |
| 5 | \( 1 + (-0.0540 - 2.23i)T \) |
| 7 | \( 1 + (1.91 - 1.82i)T \) |
good | 2 | \( 1 + (2.35 + 0.631i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (-3.08 - 1.77i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.28 + 1.28i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.792 + 2.95i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.331 + 0.191i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.658 + 2.45i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 5.51T + 29T^{2} \) |
| 31 | \( 1 + (-0.323 + 0.561i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.34 - 5.00i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (0.335 - 0.335i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.80 + 0.751i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.04 + 0.815i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.81 - 6.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.45 + 9.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 + 3.31i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.06iT - 71T^{2} \) |
| 73 | \( 1 + (0.849 + 3.17i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.21 - 1.85i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.51 + 2.63i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.3 + 10.3i)T + 97iT^{2} \) |
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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
−13.91422742785650171137322698058, −12.49578140592207992105511747324, −11.40858744101058397166799426731, −10.20700611484692793655827724922, −9.470671943456809656703646316334, −8.698856052872973090950383450335, −7.42812505936811895415784659701, −6.60093349185068136527334130359, −3.36577378315435895535545116192, −2.18424534059329627715729261420,
1.40907799761246195810057222594, 3.88764638450823581595033456217, 6.29910882516856325165866640480, 7.47313396260684716123764095687, 8.508655490968305911875606659848, 9.222048313882014623761031052898, 9.909477128415759657612519293201, 11.20367231972980237326111410798, 12.82439251279709891260277506697, 13.79525054942047429691038818292