L(s) = 1 | + (−0.892 − 1.09i)2-s + (−0.406 + 1.95i)4-s − 2.56i·5-s − i·7-s + (2.51 − 1.30i)8-s + (−2.81 + 2.28i)10-s − 1.15·11-s − 0.578·13-s + (−1.09 + 0.892i)14-s + (−3.66 − 1.59i)16-s − 5.39i·17-s − 6.20i·19-s + (5.02 + 1.04i)20-s + (1.02 + 1.26i)22-s − 7.62·23-s + ⋯ |
L(s) = 1 | + (−0.631 − 0.775i)2-s + (−0.203 + 0.979i)4-s − 1.14i·5-s − 0.377i·7-s + (0.887 − 0.460i)8-s + (−0.889 + 0.723i)10-s − 0.346·11-s − 0.160·13-s + (−0.293 + 0.238i)14-s + (−0.917 − 0.398i)16-s − 1.30i·17-s − 1.42i·19-s + (1.12 + 0.233i)20-s + (0.218 + 0.269i)22-s − 1.58·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.279811 - 0.710347i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.279811 - 0.710347i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.892 + 1.09i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.56iT - 5T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 + 0.578T + 13T^{2} \) |
| 17 | \( 1 + 5.39iT - 17T^{2} \) |
| 19 | \( 1 + 6.20iT - 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.04iT - 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 6.21iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 4.83T + 59T^{2} \) |
| 61 | \( 1 - 0.951T + 61T^{2} \) |
| 67 | \( 1 - 2.78iT - 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 8.77T + 83T^{2} \) |
| 89 | \( 1 + 5.68iT - 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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
−11.70831273820840776963384001962, −10.71382119675457153979932318716, −9.658249789168764661586911476314, −8.971671880009306514037426969254, −8.013108835046237765450748557193, −7.01839081799872343721668702579, −5.15142075707400089600373672219, −4.19806569968705283725034076764, −2.55462770732690168202472549671, −0.75314631412963245428623364912,
2.18028689047924969674911177403, 4.02359551114151119590882145478, 5.80718517266144627006626838359, 6.31624191690932900282835707897, 7.65851324407021797879170679839, 8.215784484992021929143950532852, 9.636675236642029641522906824143, 10.32161943610634545824896677881, 11.11728526141608120681896955687, 12.34503657014930292941272117361