| L(s) = 1 | + 2-s + 3-s + 4-s + 4.27·5-s + 6-s + 7-s + 8-s + 9-s + 4.27·10-s − 4·11-s + 12-s + 14-s + 4.27·15-s + 16-s + 3·17-s + 18-s + 6.54·19-s + 4.27·20-s + 21-s − 4·22-s − 5.27·23-s + 24-s + 13.2·25-s + 27-s + 28-s + 8.27·29-s + 4.27·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.91·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.35·10-s − 1.20·11-s + 0.288·12-s + 0.267·14-s + 1.10·15-s + 0.250·16-s + 0.727·17-s + 0.235·18-s + 1.50·19-s + 0.955·20-s + 0.218·21-s − 0.852·22-s − 1.09·23-s + 0.204·24-s + 2.65·25-s + 0.192·27-s + 0.188·28-s + 1.53·29-s + 0.780·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.449839666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.449839666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4.27T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 6.54T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 - 0.274T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 + 3.54T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 9T + 61T^{2} \) |
| 67 | \( 1 + 2.72T + 67T^{2} \) |
| 71 | \( 1 - 5.27T + 71T^{2} \) |
| 73 | \( 1 - 5.72T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 0.725T + 89T^{2} \) |
| 97 | \( 1 + 0.549T + 97T^{2} \) |
| 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
−7.892995120006764911377316148029, −7.21404538809100026246293232314, −6.27819956789706117867435524671, −5.78686089034513002971369049080, −5.09773136880560806908916651703, −4.64745250715409870810523125434, −3.22728870322410690888514554135, −2.81888771330621440530551957120, −1.96219388818801116165422882335, −1.26429915467338923703971919015,
1.26429915467338923703971919015, 1.96219388818801116165422882335, 2.81888771330621440530551957120, 3.22728870322410690888514554135, 4.64745250715409870810523125434, 5.09773136880560806908916651703, 5.78686089034513002971369049080, 6.27819956789706117867435524671, 7.21404538809100026246293232314, 7.892995120006764911377316148029
