L(s) = 1 | + (1.84 − 1.84i)3-s + (2.41 + 2.41i)5-s − 1.53i·7-s − 3.82i·9-s + (−3.37 − 3.37i)11-s + (0.414 − 0.414i)13-s + 8.92·15-s + 2.82·17-s + (0.317 − 0.317i)19-s + (−2.82 − 2.82i)21-s + 5.86i·23-s + 6.65i·25-s + (−1.53 − 1.53i)27-s + (−3.24 + 3.24i)29-s + 7.39·31-s + ⋯ |
L(s) = 1 | + (1.06 − 1.06i)3-s + (1.07 + 1.07i)5-s − 0.578i·7-s − 1.27i·9-s + (−1.01 − 1.01i)11-s + (0.114 − 0.114i)13-s + 2.30·15-s + 0.685·17-s + (0.0727 − 0.0727i)19-s + (−0.617 − 0.617i)21-s + 1.22i·23-s + 1.33i·25-s + (−0.294 − 0.294i)27-s + (−0.602 + 0.602i)29-s + 1.32·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.11205 - 0.874839i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.11205 - 0.874839i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1.84 + 1.84i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.53iT - 7T^{2} \) |
| 11 | \( 1 + (3.37 + 3.37i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.414 + 0.414i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + (-0.317 + 0.317i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.86iT - 23T^{2} \) |
| 29 | \( 1 + (3.24 - 3.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 + (3.58 + 3.58i)T + 37iT^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + (1.84 + 1.84i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + (5.24 + 5.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.84 + 1.84i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.24 - 9.24i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.07 - 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 + (2.48 - 2.48i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.828iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−10.60133042273829085990092265662, −9.977978395772001444740741766578, −8.891812622466073261497008566234, −7.891584474528461407573362145097, −7.29940106536115718693571368431, −6.35946214521999873930923891196, −5.45145363565279620900635943970, −3.35455143084992229620120561989, −2.76122460026501772446708132863, −1.50307546148275386429436592195,
1.95666886106626110071971252243, 2.95660436528853061084817827584, 4.52100256636222113202184407758, 5.04502765193822303228489332363, 6.15359392522684595643942538516, 7.82254255919047276277555132864, 8.577005000644167465094694793582, 9.309201589082984099557934053831, 9.928334294965333555742261260004, 10.48050516469890782476421576160