L(s) = 1 | + 2i·2-s + (1 + i)3-s − 3·4-s + (−2 + 2i)6-s + (−1 + i)7-s − 4i·8-s + i·9-s + (−3 − 3i)12-s + (−2 − 2i)14-s + 5·16-s + 17-s − 2·18-s − 2·21-s + (4 − 4i)24-s + i·25-s + ⋯ |
L(s) = 1 | + 2i·2-s + (1 + i)3-s − 3·4-s + (−2 + 2i)6-s + (−1 + i)7-s − 4i·8-s + i·9-s + (−3 − 3i)12-s + (−2 − 2i)14-s + 5·16-s + 17-s − 2·18-s − 2·21-s + (4 − 4i)24-s + i·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9542886114\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9542886114\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 - T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 - 2iT - T^{2} \) |
| 3 | \( 1 + (-1 - i)T + iT^{2} \) |
| 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (1 + i)T + iT^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 2iT - T^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1 - i)T + iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-1 + i)T - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−10.30813916166489070980551840416, −9.594040985757889731963605525407, −9.149004211195580521395692797172, −8.517647138644809170150742433872, −7.67952333185749838503850381614, −6.70849618064936116204528590558, −5.74635104893792175643103972437, −5.08622314534893812886407372604, −3.86665528610201218482511266896, −3.19389671564216599440055642828,
0.974320373461786274110774279262, 2.16326964126628773584340625955, 3.20984728687798741128797199671, 3.71067770523075155581245407326, 5.00742648746650945834870031272, 6.56318741009394828702748829611, 7.80485911257533043226855360494, 8.386354777601009821054263059587, 9.385843929804427785077256885783, 10.02162990233791658777688878230