L(s) = 1 | + i·3-s + i·7-s − 9-s − 6·11-s − 5i·13-s − 6i·17-s + 5·19-s − 21-s − 6i·23-s − i·27-s + 6·29-s + 31-s − 6i·33-s + 2i·37-s + 5·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.80·11-s − 1.38i·13-s − 1.45i·17-s + 1.14·19-s − 0.218·21-s − 1.25i·23-s − 0.192i·27-s + 1.11·29-s + 0.179·31-s − 1.04i·33-s + 0.328i·37-s + 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.085277432\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085277432\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−9.742529921245462153067943482561, −8.794207683349364235624498809072, −7.967030258758887959038005237918, −7.36044079011193432919257285284, −6.01183768748722130748896607551, −5.19477180863916879687016083153, −4.72373992798063128931570292863, −3.01295475985222491424249264213, −2.72711960451348385144025008890, −0.47598048900205289374967925284,
1.39129501132030225555070081083, 2.54407075065313153693300716550, 3.69177777094725348577589227745, 4.84624624232527391956809797003, 5.72557191323100581696881148215, 6.64003420502952692194388130617, 7.56017366500586457688626847357, 8.027814440861803343804966104436, 9.026110966715894208987583814102, 9.959717629901749504190177564788