L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 0.999·15-s + (0.499 + 0.866i)21-s + 0.999·27-s + 29-s + (−0.5 + 0.866i)31-s + (0.499 + 0.866i)33-s − 0.999·35-s + (−0.499 + 0.866i)45-s + (−0.499 − 0.866i)49-s + (−0.5 + 0.866i)53-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 0.999·15-s + (0.499 + 0.866i)21-s + 0.999·27-s + 29-s + (−0.5 + 0.866i)31-s + (0.499 + 0.866i)33-s − 0.999·35-s + (−0.499 + 0.866i)45-s + (−0.499 − 0.866i)49-s + (−0.5 + 0.866i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7581926398\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7581926398\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.70073126480046450237190066788, −9.883583031263375174942489316694, −8.802826805526790771060949719812, −8.328915675856932775456466862810, −7.08589775662895294288499797197, −6.00106889817306809408986297823, −4.92983018325448709752031177170, −4.28444397760405380763367379383, −3.35025979848401591248573946837, −0.997713551704961223929152415114,
1.80434783672156620923385269613, 2.90533663940179130952613923919, 4.43812409956674834889354206479, 5.53234022831618700437095351706, 6.48932399175406306709175052839, 7.22706844495125827639929926091, 7.982592131622134851783817364615, 8.933107251918218814393240957626, 10.08841910434541675615845718302, 11.07881001987800070929954829119