L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)25-s + 2·29-s + (−0.499 + 0.866i)32-s + (1 + 1.73i)37-s − 0.999·50-s + (1 − 1.73i)53-s + (−1 − 1.73i)58-s + 0.999·64-s + (0.999 − 1.73i)74-s + (0.499 + 0.866i)100-s − 1.99·106-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 0.999·8-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)25-s + 2·29-s + (−0.499 + 0.866i)32-s + (1 + 1.73i)37-s − 0.999·50-s + (1 − 1.73i)53-s + (−1 − 1.73i)58-s + 0.999·64-s + (0.999 − 1.73i)74-s + (0.499 + 0.866i)100-s − 1.99·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8596814840\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8596814840\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.570130037968544630607048077621, −8.449459910228350359406062402185, −8.257799438776743193375902765742, −7.09284282201505262503574200672, −6.32711052562153985071805056400, −4.99206306610946454868057275702, −4.31603067362702009851691691874, −3.19654849378673287674732862563, −2.39031932236456573789097631314, −1.04314858695646494344930395414,
1.10968720498954132696863325158, 2.57919268856991777045195546504, 3.99254377228003626569963720388, 4.87277509736667880743245682093, 5.72147085061260771859413813266, 6.49732187896102993517176655907, 7.29061222504411403034820649621, 7.961886229235568544866516932729, 8.843395094837522801935662493215, 9.337581459673687363659398076623