L(s) = 1 | + (0.981 − 0.189i)2-s + (−0.738 + 1.61i)3-s + (0.928 − 0.371i)4-s + (−0.419 + 1.72i)6-s + (0.841 − 0.540i)8-s + (−1.41 − 1.63i)9-s + (0.396 + 1.63i)11-s + (−0.0845 + 1.77i)12-s + (0.723 − 0.690i)16-s + (−1.45 − 0.584i)17-s + (−1.69 − 1.33i)18-s + (−0.473 − 1.36i)19-s + (0.698 + 1.53i)22-s + (0.252 + 1.75i)24-s + (0.841 + 0.540i)25-s + ⋯ |
L(s) = 1 | + (0.981 − 0.189i)2-s + (−0.738 + 1.61i)3-s + (0.928 − 0.371i)4-s + (−0.419 + 1.72i)6-s + (0.841 − 0.540i)8-s + (−1.41 − 1.63i)9-s + (0.396 + 1.63i)11-s + (−0.0845 + 1.77i)12-s + (0.723 − 0.690i)16-s + (−1.45 − 0.584i)17-s + (−1.69 − 1.33i)18-s + (−0.473 − 1.36i)19-s + (0.698 + 1.53i)22-s + (0.252 + 1.75i)24-s + (0.841 + 0.540i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.249934363\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.249934363\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.981 + 0.189i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
good | 3 | \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 11 | \( 1 + (-0.396 - 1.63i)T + (-0.888 + 0.458i)T^{2} \) |
| 13 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 17 | \( 1 + (1.45 + 0.584i)T + (0.723 + 0.690i)T^{2} \) |
| 19 | \( 1 + (0.473 + 1.36i)T + (-0.786 + 0.618i)T^{2} \) |
| 23 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.580 - 0.814i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.02 + 0.809i)T + (0.235 - 0.971i)T^{2} \) |
| 43 | \( 1 + (0.0671 + 0.466i)T + (-0.959 + 0.281i)T^{2} \) |
| 47 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 53 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 59 | \( 1 + (-0.0800 + 0.0514i)T + (0.415 - 0.909i)T^{2} \) |
| 61 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 71 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 73 | \( 1 + (-0.437 + 1.80i)T + (-0.888 - 0.458i)T^{2} \) |
| 79 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 83 | \( 1 + (1.38 - 1.32i)T + (0.0475 - 0.998i)T^{2} \) |
| 89 | \( 1 + (-0.815 - 1.78i)T + (-0.654 + 0.755i)T^{2} \) |
| 97 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−11.07922149878781202204762459132, −10.64745336522849978817160238266, −9.607620078217792636576464374608, −9.050770807495384995925705645596, −7.11446429895121615543777434349, −6.44312090483078982774684987663, −5.08727093125525148719460057576, −4.68281371216619557712654891305, −3.89662600556954586727333919958, −2.47084844526413006591070800537,
1.53947678530118681467843482374, 2.86375611488258122159026035960, 4.33402561726446339729311037965, 5.77999220609451254213893816738, 6.18236824919423385704008202367, 6.88723198044754964805787191818, 8.012834273645484378686692653566, 8.580948081918716184188059735206, 10.63619638814011157114354690116, 11.24377525629143485462916570108