L(s) = 1 | + (1.34 + 1.34i)2-s + (0.178 + 0.431i)3-s + 2.61i·4-s + (−0.339 + 0.820i)6-s + (−1.40 − 0.581i)7-s + (−2.17 + 2.17i)8-s + (0.552 − 0.552i)9-s + (−1.12 + 0.467i)12-s + (−1.10 − 2.67i)14-s − 3.23·16-s + (0.809 + 0.587i)17-s + 1.48·18-s − 0.710i·21-s + (−1.32 − 0.549i)24-s + (0.707 − 0.707i)25-s + ⋯ |
L(s) = 1 | + (1.34 + 1.34i)2-s + (0.178 + 0.431i)3-s + 2.61i·4-s + (−0.339 + 0.820i)6-s + (−1.40 − 0.581i)7-s + (−2.17 + 2.17i)8-s + (0.552 − 0.552i)9-s + (−1.12 + 0.467i)12-s + (−1.10 − 2.67i)14-s − 3.23·16-s + (0.809 + 0.587i)17-s + 1.48·18-s − 0.710i·21-s + (−1.32 − 0.549i)24-s + (0.707 − 0.707i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.791149651\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.791149651\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-0.809 - 0.587i)T \) |
| 47 | \( 1 - iT \) |
good | 2 | \( 1 + (-1.34 - 1.34i)T + iT^{2} \) |
| 3 | \( 1 + (-0.178 - 0.431i)T + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (1.40 + 0.581i)T + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.744 + 1.79i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1.26 + 1.26i)T + iT^{2} \) |
| 59 | \( 1 + (0.437 - 0.437i)T - iT^{2} \) |
| 61 | \( 1 + (-0.144 - 0.0600i)T + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.652 + 1.57i)T + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.652 - 1.57i)T + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 + (1 + i)T + iT^{2} \) |
| 89 | \( 1 - 1.90iT - T^{2} \) |
| 97 | \( 1 + (1.84 - 0.763i)T + (0.707 - 0.707i)T^{2} \) |
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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
−10.76351824808709325283824946112, −9.795047254867594086338887362558, −8.964745358251315492285173999695, −7.893382040596508706037848038904, −6.99652813505804169742255354611, −6.46335502430067702527733042729, −5.62257531571421170729725710615, −4.44843438381123805269516216009, −3.71907855731235172943011023353, −3.06285533267122985110171892950,
1.49259719282995548490097771660, 2.79135870140498464984674385396, 3.30827516886163669637572293290, 4.59435466099277358246153828577, 5.46698709075936681319563391281, 6.36882467384306850479505177358, 7.21656826530745335834952435561, 8.784605335479875652774326105159, 9.829195469556129938369046994322, 10.14977220069250446768653361901