| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 4·7-s + 3·8-s + 9-s + 3·11-s − 12-s − 4·14-s − 16-s − 18-s + 19-s + 4·21-s − 3·22-s + 23-s + 3·24-s + 27-s − 4·28-s − 5·29-s + 9·31-s − 5·32-s + 3·33-s − 36-s − 2·37-s − 38-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 1.06·14-s − 1/4·16-s − 0.235·18-s + 0.229·19-s + 0.872·21-s − 0.639·22-s + 0.208·23-s + 0.612·24-s + 0.192·27-s − 0.755·28-s − 0.928·29-s + 1.61·31-s − 0.883·32-s + 0.522·33-s − 1/6·36-s − 0.328·37-s − 0.162·38-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613197153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613197153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.375586930532197171435102084043, −8.721157197040644373439868435028, −8.059132773379975443514972561189, −7.56262888744563490820286573499, −6.46872919351205495550038689249, −5.09570305472347318169914513335, −4.52175123062198989941702185993, −3.54975381370698875378500028111, −1.98349098766325641228995661317, −1.11188483392532769774032160005,
1.11188483392532769774032160005, 1.98349098766325641228995661317, 3.54975381370698875378500028111, 4.52175123062198989941702185993, 5.09570305472347318169914513335, 6.46872919351205495550038689249, 7.56262888744563490820286573499, 8.059132773379975443514972561189, 8.721157197040644373439868435028, 9.375586930532197171435102084043
