L(s) = 1 | + 2-s + (1.5 − 0.866i)3-s + 4-s − 1.73i·5-s + (1.5 − 0.866i)6-s + (−2.5 + 0.866i)7-s + 8-s + (1.5 − 2.59i)9-s − 1.73i·10-s + (1.5 − 0.866i)12-s + (1 − 3.46i)13-s + (−2.5 + 0.866i)14-s + (−1.49 − 2.59i)15-s + 16-s + 3·17-s + (1.5 − 2.59i)18-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.866 − 0.499i)3-s + 0.5·4-s − 0.774i·5-s + (0.612 − 0.353i)6-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s − 0.547i·10-s + (0.433 − 0.249i)12-s + (0.277 − 0.960i)13-s + (−0.668 + 0.231i)14-s + (−0.387 − 0.670i)15-s + 0.250·16-s + 0.727·17-s + (0.353 − 0.612i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.38878 - 1.29688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.38878 - 1.29688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 1.73iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 13.8iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + 3.46iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.64699407848347568329041107502, −9.593923597260431743651513723146, −8.905848056954869280083780155421, −7.912272905360040515456890246534, −7.07115744253702890159614085922, −5.97424731968802049408636226979, −5.09740903242519650302606083236, −3.58850534674695920434636068960, −2.98194318004286500581533834895, −1.34187722864537857370977683860,
2.20195601855288858272551951258, 3.35154370495190499162848631191, 3.90665101802109905137323686306, 5.21182945760726672695674910454, 6.53349746314984434834244872719, 7.12696457284613766095951818614, 8.224239765567810305698716476228, 9.389059942041548515016903396601, 10.06353085932936194211243375785, 10.84509625128243547410197959699