| L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 9-s − 4·13-s − 2·15-s − 12·17-s − 8·19-s + 8·21-s + 25-s + 4·27-s − 12·29-s − 4·35-s − 4·37-s + 8·39-s + 45-s − 2·49-s + 24·51-s + 16·57-s − 4·63-s − 4·65-s + 24·71-s − 2·75-s − 11·81-s + 12·83-s − 12·85-s + 24·87-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s − 2.91·17-s − 1.83·19-s + 1.74·21-s + 1/5·25-s + 0.769·27-s − 2.22·29-s − 0.676·35-s − 0.657·37-s + 1.28·39-s + 0.149·45-s − 2/7·49-s + 3.36·51-s + 2.11·57-s − 0.503·63-s − 0.496·65-s + 2.84·71-s − 0.230·75-s − 1.22·81-s + 1.31·83-s − 1.30·85-s + 2.57·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.404829898294941223649273971532, −8.008294857926527437201173134092, −6.96167091691043492420755836355, −6.87772159023272676690262517558, −6.41842389830764202453884824049, −6.26903409609752801893873868553, −5.32703134203753576211066117067, −5.24854435453179347418331807133, −4.28530365432405873873397666950, −4.13045861856120535222472022668, −3.16807967427295861033734664069, −2.32736297473362010207341859714, −2.01836233216080453835298575952, 0, 0,
2.01836233216080453835298575952, 2.32736297473362010207341859714, 3.16807967427295861033734664069, 4.13045861856120535222472022668, 4.28530365432405873873397666950, 5.24854435453179347418331807133, 5.32703134203753576211066117067, 6.26903409609752801893873868553, 6.41842389830764202453884824049, 6.87772159023272676690262517558, 6.96167091691043492420755836355, 8.008294857926527437201173134092, 8.404829898294941223649273971532
