L(s) = 1 | + 2-s + (0.5 + 0.866i)7-s − 8-s − 9-s − 11-s + (0.5 + 0.866i)14-s − 16-s − 18-s + 1.73i·19-s − 22-s + 23-s − 25-s − 29-s + 1.73i·31-s + 1.73i·38-s + ⋯ |
L(s) = 1 | + 2-s + (0.5 + 0.866i)7-s − 8-s − 9-s − 11-s + (0.5 + 0.866i)14-s − 16-s − 18-s + 1.73i·19-s − 22-s + 23-s − 25-s − 29-s + 1.73i·31-s + 1.73i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.053231845\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053231845\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - 1.73iT - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.73iT - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + 1.73iT - T^{2} \) |
| 61 | \( 1 + 1.73iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.73iT - T^{2} \) |
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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
−9.068185897651109290202133155642, −8.165212333626475066163691916678, −7.82379562498106380316534201331, −6.43539811039768314637720029380, −5.72759633888322940511678949728, −5.33677324405041114899652064935, −4.63504284814224181809837441960, −3.48091883111733804671273564075, −2.91906473359937282052904275909, −1.87545506818864344996316884521,
0.42002563221972887963389766486, 2.33921327199187931071240965184, 3.05211325529967211730292776187, 4.04142732529596503309845800930, 4.69378516200538305512138604879, 5.45498497542622004492528608558, 5.95656712289839372661099339297, 7.11147937566692854222703472103, 7.66046299251959614395475364462, 8.616393398478971506303617956070