| L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s − 6·13-s + 2·15-s − 6·17-s − 8·19-s − 4·21-s − 25-s − 27-s + 6·29-s − 8·35-s + 6·37-s + 6·39-s + 10·41-s − 8·43-s − 2·45-s + 9·49-s + 6·51-s + 6·53-s + 8·57-s − 4·59-s + 2·61-s + 4·63-s + 12·65-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.35·35-s + 0.986·37-s + 0.960·39-s + 1.56·41-s − 1.21·43-s − 0.298·45-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 1.05·57-s − 0.520·59-s + 0.256·61-s + 0.503·63-s + 1.48·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9275525465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9275525465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.228415179514193715605028768454, −7.35824269240573348608620103444, −6.84172401965691206022768509346, −5.96726719550806915467982642379, −4.94084641236370924155494583195, −4.49914350038850325136622917050, −4.12248457413213219306418931774, −2.54287436292890018722810885401, −1.94546966374609697836921373063, −0.50708313336612890338490237295,
0.50708313336612890338490237295, 1.94546966374609697836921373063, 2.54287436292890018722810885401, 4.12248457413213219306418931774, 4.49914350038850325136622917050, 4.94084641236370924155494583195, 5.96726719550806915467982642379, 6.84172401965691206022768509346, 7.35824269240573348608620103444, 8.228415179514193715605028768454
