| L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s − 7-s − 3·8-s + 9-s + 10-s + 12-s − 14-s − 15-s − 16-s − 17-s + 18-s + 2·19-s − 20-s + 21-s + 6·23-s + 3·24-s + 25-s − 27-s + 28-s − 4·29-s − 30-s − 4·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s + 1.25·23-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.742·29-s − 0.182·30-s − 0.718·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609357995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609357995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.374382510909813445824217205848, −8.702357198738553963201149941811, −7.50835306182715923625670381327, −6.67931663262518992670154334973, −5.79530321478153998743024153736, −5.32363110257955187072323371443, −4.40788901588214039441553602026, −3.55589293847504590895677387951, −2.47789863858984368696646328047, −0.809464007514526215837722664467,
0.809464007514526215837722664467, 2.47789863858984368696646328047, 3.55589293847504590895677387951, 4.40788901588214039441553602026, 5.32363110257955187072323371443, 5.79530321478153998743024153736, 6.67931663262518992670154334973, 7.50835306182715923625670381327, 8.702357198738553963201149941811, 9.374382510909813445824217205848
