L(s) = 1 | − 2·3-s − 6·5-s + 3·9-s + 11-s + 12·15-s − 4·16-s − 8·23-s + 17·25-s − 4·27-s − 12·31-s − 2·33-s − 18·45-s − 18·47-s + 8·48-s + 11·49-s + 20·53-s − 6·55-s − 16·59-s + 16·67-s + 16·69-s − 24·71-s − 34·75-s + 24·80-s + 5·81-s − 12·89-s + 24·93-s − 20·97-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 2.68·5-s + 9-s + 0.301·11-s + 3.09·15-s − 16-s − 1.66·23-s + 17/5·25-s − 0.769·27-s − 2.15·31-s − 0.348·33-s − 2.68·45-s − 2.62·47-s + 1.15·48-s + 11/7·49-s + 2.74·53-s − 0.809·55-s − 2.08·59-s + 1.95·67-s + 1.92·69-s − 2.84·71-s − 3.92·75-s + 2.68·80-s + 5/9·81-s − 1.27·89-s + 2.48·93-s − 2.03·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.114016907984113880616472369357, −7.47848778675296589128891353641, −7.43755268339255182270866775688, −6.95333653112949028642183365593, −6.41173993136661837405396202601, −5.82201250996795631397735521238, −5.30575361876571555675709307141, −4.64831433194343792963963650578, −4.23023339125662356328251569037, −3.81591641511052084615077615666, −3.57553622245677274525627107284, −2.47849673392805399692938216213, −1.44645188395830487846696686994, 0, 0,
1.44645188395830487846696686994, 2.47849673392805399692938216213, 3.57553622245677274525627107284, 3.81591641511052084615077615666, 4.23023339125662356328251569037, 4.64831433194343792963963650578, 5.30575361876571555675709307141, 5.82201250996795631397735521238, 6.41173993136661837405396202601, 6.95333653112949028642183365593, 7.43755268339255182270866775688, 7.47848778675296589128891353641, 8.114016907984113880616472369357