L(s) = 1 | + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.347 + 1.52i)5-s + (0.781 + 0.623i)8-s + (−0.678 + 1.40i)10-s + (−1.12 − 1.40i)13-s + (0.623 + 0.781i)16-s + 0.445i·17-s + (−0.974 + 1.22i)20-s + (−1.30 − 0.626i)25-s + (−0.781 − 1.62i)26-s + (0.781 − 0.623i)29-s + (0.433 + 0.900i)32-s + (−0.0990 + 0.433i)34-s + (0.678 + 0.541i)37-s + ⋯ |
L(s) = 1 | + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.347 + 1.52i)5-s + (0.781 + 0.623i)8-s + (−0.678 + 1.40i)10-s + (−1.12 − 1.40i)13-s + (0.623 + 0.781i)16-s + 0.445i·17-s + (−0.974 + 1.22i)20-s + (−1.30 − 0.626i)25-s + (−0.781 − 1.62i)26-s + (0.781 − 0.623i)29-s + (0.433 + 0.900i)32-s + (−0.0990 + 0.433i)34-s + (0.678 + 0.541i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1044 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.710220509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710220509\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.974 - 0.222i)T \) |
| 3 | \( 1 \) |
| 29 | \( 1 + (-0.781 + 0.623i)T \) |
good | 5 | \( 1 + (0.347 - 1.52i)T + (-0.900 - 0.433i)T^{2} \) |
| 7 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 13 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 - 0.445iT - T^{2} \) |
| 19 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 23 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 37 | \( 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + 1.80iT - T^{2} \) |
| 43 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 53 | \( 1 + (0.433 - 1.90i)T + (-0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-1.90 + 0.433i)T + (0.900 - 0.433i)T^{2} \) |
| 79 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (1.21 + 0.277i)T + (0.900 + 0.433i)T^{2} \) |
| 97 | \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \) |
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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
−10.63577189377436587012031778904, −9.764806084517464610459752913886, −8.156003057635019552538162276414, −7.59282655005251210284455522953, −6.85418502199538962752209053403, −6.06488850199766220937593387308, −5.15340458715805997395942189704, −4.00209191345660694751305716368, −3.06364718604947944603517630796, −2.40904022631874351370430049965,
1.35904443797505262076684347848, 2.62420663887542020562666385464, 4.07915957578496144031964426983, 4.67995724382273683678476513736, 5.29074508596784750388741762704, 6.47582278754217188732405485820, 7.33098846375533319147825961928, 8.276495829300791232279027759366, 9.302066298131165284928384281435, 9.834431668557275836982822310787