L(s) = 1 | − i·3-s − 2i·5-s − 7-s − 9-s + 6i·13-s − 2·15-s − 2·17-s + 4i·19-s + i·21-s + 4·23-s + 25-s + i·27-s − 10i·29-s − 8·31-s + 2i·35-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.894i·5-s − 0.377·7-s − 0.333·9-s + 1.66i·13-s − 0.516·15-s − 0.485·17-s + 0.917i·19-s + 0.218i·21-s + 0.834·23-s + 0.200·25-s + 0.192i·27-s − 1.85i·29-s − 1.43·31-s + 0.338i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.641554971\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.641554971\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.070828521745769433905262132898, −7.36006052781597296038361986784, −6.67065331230782715426301881180, −6.00865317510468795078881895421, −5.24487690905582453169888251925, −4.32121149305460328474051728124, −3.78826539982888105797623999812, −2.45747369397515860120810417977, −1.74755006550206089763801434239, −0.67863994464706470924167234095,
0.68741887355378035962516818628, 2.28598386232321865461817933523, 3.24336431697106323907411110975, 3.39315890964613913581494508387, 4.73676751849294263269988136785, 5.30884827598685832383569869188, 6.08053601902366725784986321666, 7.03156955045598892915018315466, 7.27951004991481243082862315161, 8.418660075096559234974314248101