L(s) = 1 | + i·3-s + 2i·7-s + 2·9-s + 3·11-s − 4i·13-s − 3i·17-s + 5·19-s − 2·21-s − 6i·23-s + 5i·27-s + 2·31-s + 3i·33-s − 2i·37-s + 4·39-s − 3·41-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.755i·7-s + 0.666·9-s + 0.904·11-s − 1.10i·13-s − 0.727i·17-s + 1.14·19-s − 0.436·21-s − 1.25i·23-s + 0.962i·27-s + 0.359·31-s + 0.522i·33-s − 0.328i·37-s + 0.640·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.996966738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.996966738\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.413122181501121067502585812993, −8.907364742296679397927542322911, −7.896582581441935673004699124551, −7.09069166000098794953437238075, −6.14190369361402019347132263409, −5.26917074365722008247656327573, −4.51635204004326823266481540614, −3.48975293134307007781769081762, −2.56412917424182407302898250736, −1.04352949765364317172325986919,
1.12105277475785692415343021852, 1.88649838030091600330961784947, 3.54741066566987494914322703675, 4.14354412034294634340637342580, 5.22013194175450092371537849598, 6.42963706496312371619093103027, 6.92206168735307911302235535747, 7.59429110607486304330216569176, 8.466732275729835641972245163985, 9.537298808825653968456963022092