| L(s) = 1 | − 3·3-s − 3·4-s − 5·7-s + 6·9-s + 9·12-s − 2·13-s + 5·16-s + 19-s + 15·21-s − 4·25-s − 9·27-s + 15·28-s − 8·31-s − 18·36-s + 37-s + 6·39-s − 4·43-s − 15·48-s + 5·49-s + 6·52-s − 3·57-s + 12·61-s − 30·63-s − 3·64-s − 8·67-s + 6·73-s + 12·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 3/2·4-s − 1.88·7-s + 2·9-s + 2.59·12-s − 0.554·13-s + 5/4·16-s + 0.229·19-s + 3.27·21-s − 4/5·25-s − 1.73·27-s + 2.83·28-s − 1.43·31-s − 3·36-s + 0.164·37-s + 0.960·39-s − 0.609·43-s − 2.16·48-s + 5/7·49-s + 0.832·52-s − 0.397·57-s + 1.53·61-s − 3.77·63-s − 3/8·64-s − 0.977·67-s + 0.702·73-s + 1.38·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2547 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 283 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.73361622282280365746979731014, −12.28453735112405363569019339066, −11.62903219050268284837873551308, −10.84319025330066361427143363918, −10.13116881758474459595062559948, −9.566265883109564650648456216951, −9.457627220941446651639972325774, −8.349764713614763396584200304982, −7.25299589451818783255641711548, −6.64062488163986775931412221451, −5.82079413706745846722911821346, −5.30126130438548452981414146530, −4.38861303129640457610369357257, −3.52128976331075074185573155388, 0,
3.52128976331075074185573155388, 4.38861303129640457610369357257, 5.30126130438548452981414146530, 5.82079413706745846722911821346, 6.64062488163986775931412221451, 7.25299589451818783255641711548, 8.349764713614763396584200304982, 9.457627220941446651639972325774, 9.566265883109564650648456216951, 10.13116881758474459595062559948, 10.84319025330066361427143363918, 11.62903219050268284837873551308, 12.28453735112405363569019339066, 12.73361622282280365746979731014
