L(s) = 1 | + (1.10 + 0.885i)2-s + (0.430 + 1.95i)4-s + 2.99i·5-s + 7-s + (−1.25 + 2.53i)8-s + (−2.64 + 3.29i)10-s + 2.13i·11-s + 0.665i·13-s + (1.10 + 0.885i)14-s + (−3.62 + 1.68i)16-s + 7.04·17-s − 1.39i·19-s + (−5.83 + 1.28i)20-s + (−1.89 + 2.35i)22-s + 0.184·23-s + ⋯ |
L(s) = 1 | + (0.779 + 0.626i)2-s + (0.215 + 0.976i)4-s + 1.33i·5-s + 0.377·7-s + (−0.443 + 0.896i)8-s + (−0.837 + 1.04i)10-s + 0.645i·11-s + 0.184i·13-s + (0.294 + 0.236i)14-s + (−0.907 + 0.420i)16-s + 1.70·17-s − 0.320i·19-s + (−1.30 + 0.287i)20-s + (−0.404 + 0.502i)22-s + 0.0384·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.623095711\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.623095711\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.10 - 0.885i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 2.99iT - 5T^{2} \) |
| 11 | \( 1 - 2.13iT - 11T^{2} \) |
| 13 | \( 1 - 0.665iT - 13T^{2} \) |
| 17 | \( 1 - 7.04T + 17T^{2} \) |
| 19 | \( 1 + 1.39iT - 19T^{2} \) |
| 23 | \( 1 - 0.184T + 23T^{2} \) |
| 29 | \( 1 - 1.27iT - 29T^{2} \) |
| 31 | \( 1 + 7.62T + 31T^{2} \) |
| 37 | \( 1 + 6.69iT - 37T^{2} \) |
| 41 | \( 1 + 0.274T + 41T^{2} \) |
| 43 | \( 1 - 2.63iT - 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 + 7.48iT - 53T^{2} \) |
| 59 | \( 1 - 9.03iT - 59T^{2} \) |
| 61 | \( 1 - 9.80iT - 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 - 7.98T + 79T^{2} \) |
| 83 | \( 1 + 6.76iT - 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 0.107T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.869663515472978139521016565633, −8.890828037333211885690385076936, −7.72165464026491167993560267621, −7.39120824251154499102360064224, −6.60993218110159278713290495448, −5.73982690098848021962811611290, −4.94914103599489286577160420263, −3.80177045803990635963566614735, −3.10641217215800797685955605617, −2.02596906695628852353008408433,
0.823915368110465629268879838629, 1.71799493217036957665879808139, 3.18222772723133568763183667197, 3.98649970290695518384638750742, 5.11256984266111985247234482826, 5.39247178831828328231474861521, 6.36165596135735590675354966716, 7.67276325871522199121844590132, 8.389631420360916557120919259107, 9.289029614498255447460692680766