L(s) = 1 | + (0.866 − 0.5i)5-s − 7-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + i·13-s + (−0.866 − 1.5i)19-s + (1.5 − 0.866i)23-s + (0.499 − 0.866i)25-s + (−0.866 + 0.5i)35-s + (−0.866 − 1.5i)37-s − 1.73i·41-s + (−0.866 − 0.499i)45-s + (0.5 + 0.866i)47-s + 49-s + (−0.866 + 1.5i)53-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)5-s − 7-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + i·13-s + (−0.866 − 1.5i)19-s + (1.5 − 0.866i)23-s + (0.499 − 0.866i)25-s + (−0.866 + 0.5i)35-s + (−0.866 − 1.5i)37-s − 1.73i·41-s + (−0.866 − 0.499i)45-s + (0.5 + 0.866i)47-s + 49-s + (−0.866 + 1.5i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9318112019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9318112019\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−8.954035764029244767383452544064, −8.756042620921700765288622334512, −7.22886038050135442489663022887, −6.59762937205670122504599416109, −5.95410677236184038427274856957, −5.13352625096269504344497212733, −4.18354205768609939675246829255, −3.00435125055949952636846801892, −2.31984046834038975128933602946, −0.60678915614507729832208699826,
1.76731829101879436936132944281, 2.82949625001868334697811586594, 3.35535425623524273575525125883, 4.91504564039602268293286456966, 5.52085159446384092804209424749, 6.25201011921830346447803686931, 7.06198278804283564482822237802, 7.912606515679479477196660813205, 8.596098460885415244496047056495, 9.695083370201902964952095888705