| L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s + 9-s + 3·10-s − 12-s − 5·13-s − 14-s + 3·15-s + 16-s + 17-s − 18-s − 4·19-s − 3·20-s − 21-s + 23-s + 24-s + 4·25-s + 5·26-s − 27-s + 28-s − 3·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s − 0.218·21-s + 0.208·23-s + 0.204·24-s + 4/5·25-s + 0.980·26-s − 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.44798950555891, −13.45580870518859, −12.76645432994983, −12.60281191295025, −11.94344945348059, −11.53157866646961, −11.32140304955447, −10.73958600420242, −10.14970753142561, −9.818761558859929, −9.163275396500945, −8.558387237455527, −8.105732264221084, −7.663564413349586, −7.237088718728857, −6.775931929840637, −6.219166531524592, −5.416467478293483, −4.949352769510105, −4.437131088145918, −3.829823182391275, −3.239957228136519, −2.456627939585857, −1.836951028315580, −1.032850247970037, 0, 0,
1.032850247970037, 1.836951028315580, 2.456627939585857, 3.239957228136519, 3.829823182391275, 4.437131088145918, 4.949352769510105, 5.416467478293483, 6.219166531524592, 6.775931929840637, 7.237088718728857, 7.663564413349586, 8.105732264221084, 8.558387237455527, 9.163275396500945, 9.818761558859929, 10.14970753142561, 10.73958600420242, 11.32140304955447, 11.53157866646961, 11.94344945348059, 12.60281191295025, 12.76645432994983, 13.45580870518859, 14.44798950555891
