L(s) = 1 | + (1.02 + 1.77i)2-s + (0.5 + 0.866i)3-s + (−1.09 + 1.90i)4-s + 3.35·5-s + (−1.02 + 1.77i)6-s + (−1.12 + 1.94i)7-s − 0.405·8-s + (−0.499 + 0.866i)9-s + (3.43 + 5.95i)10-s + (−2.46 − 4.27i)11-s − 2.19·12-s − 4.60·14-s + (1.67 + 2.90i)15-s + (1.78 + 3.08i)16-s + (−0.455 + 0.789i)17-s − 2.04·18-s + ⋯ |
L(s) = 1 | + (0.724 + 1.25i)2-s + (0.288 + 0.499i)3-s + (−0.549 + 0.951i)4-s + 1.50·5-s + (−0.418 + 0.724i)6-s + (−0.424 + 0.735i)7-s − 0.143·8-s + (−0.166 + 0.288i)9-s + (1.08 + 1.88i)10-s + (−0.744 − 1.28i)11-s − 0.634·12-s − 1.23·14-s + (0.433 + 0.750i)15-s + (0.445 + 0.771i)16-s + (−0.110 + 0.191i)17-s − 0.482·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25119 + 2.36616i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25119 + 2.36616i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-1.02 - 1.77i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + (1.12 - 1.94i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.46 + 4.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.455 - 0.789i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.90 + 3.29i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.01 + 1.75i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.96 - 3.41i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 8.82T + 31T^{2} \) |
| 37 | \( 1 + (4.40 + 7.62i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.46 + 6.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.14 + 1.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 - 0.542T + 53T^{2} \) |
| 59 | \( 1 + (-2.35 + 4.08i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.83 - 3.18i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.760 - 1.31i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.18 - 2.05i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 2.30T + 83T^{2} \) |
| 89 | \( 1 + (-5.02 - 8.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.06 + 13.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.94566125643218931045478215651, −10.29676532875903782913572508391, −9.090956536021947779279112073407, −8.686171390478885916671202472515, −7.36388976512017438375324763397, −6.28948616808406321870993245873, −5.56707934906715591583436472021, −5.17607176035445182224520503183, −3.55375818503327849024939969543, −2.31739845005765211038363744270,
1.53504347334447203548324367310, 2.32161877120366555697115602025, 3.44348680073405723937489088509, 4.73937254203500709960744186525, 5.68667384737958148988396868539, 6.88060921120070803746993161214, 7.77189529993606639231076936315, 9.355875033640024465656369038901, 10.05731846616667382092406352121, 10.39842787228495702583072143242