L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)19-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)28-s − 31-s + 0.999·36-s + 37-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)19-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.499 − 0.866i)28-s − 31-s + 0.999·36-s + 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4872960951\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4872960951\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 - T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−13.62304570253849326090067990079, −12.87351548979488832581956316560, −11.50359384581484360434866422988, −10.96748126909624518403723367049, −9.323550309234632306437585417334, −8.463172696586380663441779432537, −6.86328204660327443830310453966, −5.81220143111909919666559615863, −4.74283713726553977917887541470, −1.94869000785363089009706569365,
3.60018671529121378340731000668, 4.48691441166618421750396295135, 5.97703246902699662443433916656, 7.70723203883669501750919805383, 8.616652078435595282895870814145, 10.00763679155022500534844766937, 10.80389228668928613087785431902, 11.95902905251912013524634114598, 12.93352522765451202314856167685, 14.08859395282092172934470765459