L(s) = 1 | + 3·3-s + 6·5-s − 2·7-s + 6·9-s + 9·13-s + 18·15-s − 6·17-s + 8·19-s − 6·21-s + 19·25-s + 9·27-s − 6·29-s − 12·35-s + 27·39-s − 12·41-s − 43-s + 36·45-s − 12·47-s − 11·49-s − 18·51-s − 12·53-s + 24·57-s − 7·61-s − 12·63-s + 54·65-s − 15·67-s + 6·71-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 2.68·5-s − 0.755·7-s + 2·9-s + 2.49·13-s + 4.64·15-s − 1.45·17-s + 1.83·19-s − 1.30·21-s + 19/5·25-s + 1.73·27-s − 1.11·29-s − 2.02·35-s + 4.32·39-s − 1.87·41-s − 0.152·43-s + 5.36·45-s − 1.75·47-s − 1.57·49-s − 2.52·51-s − 1.64·53-s + 3.17·57-s − 0.896·61-s − 1.51·63-s + 6.69·65-s − 1.83·67-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.092960273\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.092960273\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 95 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.928233454302836245203350097863, −9.669219937949775089496725376886, −9.595050222532212835509216804470, −9.095254492352062399537752274407, −8.670108693975763968142596658673, −8.584785539635644445243430999637, −7.88193314646383327811318753797, −7.37647689978811065529997490085, −6.61508926834439222043164833043, −6.55343454635500180073030035466, −6.00499403050244054769062561264, −5.78190990133850197054414974308, −4.95657816502970387590339440363, −4.63733297448008114901755583472, −3.51084002464723050754903134445, −3.37667335065790969181971061924, −3.01652075159725567265311669795, −2.11295748302503760662776943399, −1.65629867996553385547471978989, −1.44884665951825694270658039340,
1.44884665951825694270658039340, 1.65629867996553385547471978989, 2.11295748302503760662776943399, 3.01652075159725567265311669795, 3.37667335065790969181971061924, 3.51084002464723050754903134445, 4.63733297448008114901755583472, 4.95657816502970387590339440363, 5.78190990133850197054414974308, 6.00499403050244054769062561264, 6.55343454635500180073030035466, 6.61508926834439222043164833043, 7.37647689978811065529997490085, 7.88193314646383327811318753797, 8.584785539635644445243430999637, 8.670108693975763968142596658673, 9.095254492352062399537752274407, 9.595050222532212835509216804470, 9.669219937949775089496725376886, 9.928233454302836245203350097863