L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + 0.618·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (−1.30 + 0.951i)11-s + (0.809 + 0.587i)12-s + (0.500 − 0.363i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s − 0.999·18-s + ⋯ |
L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)6-s + 0.618·7-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (−1.30 + 0.951i)11-s + (0.809 + 0.587i)12-s + (0.500 − 0.363i)14-s + (0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s − 0.999·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.378112086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378112086\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
good | 7 | \( 1 - 0.618T + T^{2} \) |
| 11 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.71161272849659185357909767531, −10.16971236124642587232353105257, −9.437987246934160172361382156219, −8.452096164272570673739495733449, −6.97670638188966200650886100836, −5.69430914124502981247248948137, −5.07288703322210171428009826633, −4.54041723275158527535589520431, −3.10520709311688864591500395430, −1.82616607877136577685655652354,
2.15081326373838919591372847816, 3.03312472714709196783661963924, 4.75093527236746108051610999977, 5.89463443287140223083459360678, 6.02856331054450554983654699819, 7.43258593177356800160634817136, 7.84232125263057793666246361886, 8.863762313301936080766506682848, 10.39695385323574811243124444611, 11.18315070121590300584834859165