L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.309 − 0.535i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.309 − 0.535i)10-s + (−0.809 + 1.40i)13-s + (−0.5 − 0.866i)16-s + (0.809 + 1.40i)17-s − 0.999·18-s + 0.618·20-s + (0.309 − 0.535i)25-s − 1.61·26-s + (−0.809 + 1.40i)29-s + (0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.309 − 0.535i)5-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (0.309 − 0.535i)10-s + (−0.809 + 1.40i)13-s + (−0.5 − 0.866i)16-s + (0.809 + 1.40i)17-s − 0.999·18-s + 0.618·20-s + (0.309 − 0.535i)25-s − 1.61·26-s + (−0.809 + 1.40i)29-s + (0.499 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.003039032\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.003039032\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.618T + T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.860175199817453215175104301307, −8.881671517773725450667870621363, −8.391783593350517773168075347773, −7.57766844167902998263631743852, −6.84327512961047451574053889254, −5.85123660608135745255920891589, −5.05760603071780430821458836895, −4.36607940666843434441230193830, −3.40030461555185866841462237874, −2.00856484657808300727281370499,
0.69174455490200203247569936909, 2.56945236829897877167242457904, 3.14636443953080875257334410212, 4.04033696433987319656180588207, 5.31234922406461895780283424625, 5.73834208868317462285961222036, 6.97130245104689900847299815515, 7.72331063133816183423015279080, 8.861731302317050473381694622000, 9.637351308929701355016244041782