| L(s) = 1 | − 2-s − 0.705·3-s + 4-s + 0.580·5-s + 0.705·6-s − 4.68·7-s − 8-s − 2.50·9-s − 0.580·10-s − 4.95·11-s − 0.705·12-s + 1.86·13-s + 4.68·14-s − 0.409·15-s + 16-s + 6.13·17-s + 2.50·18-s + 5.02·19-s + 0.580·20-s + 3.30·21-s + 4.95·22-s + 1.58·23-s + 0.705·24-s − 4.66·25-s − 1.86·26-s + 3.88·27-s − 4.68·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.407·3-s + 0.5·4-s + 0.259·5-s + 0.287·6-s − 1.77·7-s − 0.353·8-s − 0.834·9-s − 0.183·10-s − 1.49·11-s − 0.203·12-s + 0.515·13-s + 1.25·14-s − 0.105·15-s + 0.250·16-s + 1.48·17-s + 0.589·18-s + 1.15·19-s + 0.129·20-s + 0.721·21-s + 1.05·22-s + 0.330·23-s + 0.143·24-s − 0.932·25-s − 0.364·26-s + 0.746·27-s − 0.885·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6347434477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6347434477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 0.705T + 3T^{2} \) |
| 5 | \( 1 - 0.580T + 5T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 - 0.121T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 3.30T + 61T^{2} \) |
| 67 | \( 1 - 0.357T + 67T^{2} \) |
| 71 | \( 1 - 5.06T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.09305017133137957179507555355, −9.581974918194055275186273259019, −8.472388505733419348611063415350, −7.73868174526617357018100207316, −6.64133837541557826828257554048, −5.92953068727667598029361917153, −5.22547004904049135923894834855, −3.26697712106623518898772594406, −2.79552848452139372375724727939, −0.69731509541020513539611564516,
0.69731509541020513539611564516, 2.79552848452139372375724727939, 3.26697712106623518898772594406, 5.22547004904049135923894834855, 5.92953068727667598029361917153, 6.64133837541557826828257554048, 7.73868174526617357018100207316, 8.472388505733419348611063415350, 9.581974918194055275186273259019, 10.09305017133137957179507555355
