| L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s + 4·17-s + 3·25-s − 8·35-s + 12·37-s − 12·41-s − 16·43-s − 6·45-s + 8·47-s + 9·49-s − 8·59-s + 12·63-s + 16·67-s + 9·81-s − 32·83-s + 8·85-s − 12·89-s + 12·101-s + 28·109-s − 16·119-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s + 0.970·17-s + 3/5·25-s − 1.35·35-s + 1.97·37-s − 1.87·41-s − 2.43·43-s − 0.894·45-s + 1.16·47-s + 9/7·49-s − 1.04·59-s + 1.51·63-s + 1.95·67-s + 81-s − 3.51·83-s + 0.867·85-s − 1.27·89-s + 1.19·101-s + 2.68·109-s − 1.46·119-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.209298532389627376088464017410, −7.68202906061508776443342009270, −6.97238544391138654785050907556, −6.73987218972262139901896750788, −6.26144985166624391247223284815, −5.70093888866808737514223874840, −5.65583754219252934917578600426, −4.95784212823950050876277151155, −4.36298506474338616056729023671, −3.47931878349805096580870899569, −3.26180587408034592610487247010, −2.72897396070188084113342251167, −2.09014243085294226746390952303, −1.14003521662233099173364940452, 0,
1.14003521662233099173364940452, 2.09014243085294226746390952303, 2.72897396070188084113342251167, 3.26180587408034592610487247010, 3.47931878349805096580870899569, 4.36298506474338616056729023671, 4.95784212823950050876277151155, 5.65583754219252934917578600426, 5.70093888866808737514223874840, 6.26144985166624391247223284815, 6.73987218972262139901896750788, 6.97238544391138654785050907556, 7.68202906061508776443342009270, 8.209298532389627376088464017410
