| L(s) = 1 | − 1.73i·5-s + 1.26·7-s − 1.26i·11-s + 3i·13-s − 4.26·17-s + 4.19i·19-s + 1.26·23-s + 2.00·25-s + 4.26i·29-s − 3.46·31-s − 2.19i·35-s + 0.464i·37-s − 3.46·41-s + 6.19i·43-s − 12.9·47-s + ⋯ |
| L(s) = 1 | − 0.774i·5-s + 0.479·7-s − 0.382i·11-s + 0.832i·13-s − 1.03·17-s + 0.962i·19-s + 0.264·23-s + 0.400·25-s + 0.792i·29-s − 0.622·31-s − 0.371i·35-s + 0.0762i·37-s − 0.541·41-s + 0.944i·43-s − 1.88·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9456609275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9456609275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.26iT - 11T^{2} \) |
| 13 | \( 1 - 3iT - 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 - 4.19iT - 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 4.26iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 0.464iT - 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 6.19iT - 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 0.928iT - 53T^{2} \) |
| 59 | \( 1 + 9.46iT - 59T^{2} \) |
| 61 | \( 1 - 6.46iT - 61T^{2} \) |
| 67 | \( 1 + 4.19iT - 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 5T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 0.803T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.372122823822904181861489321610, −7.940448916113374550440594375719, −6.85204240352345257746983055860, −6.39011488384870668794015493597, −5.31871271597306021886102280585, −4.82926431375109203657505632617, −4.07231495013921461146069993860, −3.17587429762282676042048294076, −1.95499664056508976699163920709, −1.25780254971609048363311155652,
0.24541452351697305752618739757, 1.72247378872722450750951894180, 2.62254584059497085429248051432, 3.32934385239527753322842380587, 4.39160275731996004107986256589, 5.01267277764484137127771885455, 5.85647632336576310331453000070, 6.80329045309389982942462667586, 7.09371811120981760246839370479, 8.025379138067926784325428880645
