L(s) = 1 | − 2·3-s − 4-s − 2·7-s + 9-s + 2·12-s + 4·13-s − 3·16-s + 4·19-s + 4·21-s + 2·25-s + 4·27-s + 2·28-s − 14·31-s − 36-s + 4·37-s − 8·39-s − 2·43-s + 6·48-s − 2·49-s − 4·52-s − 8·57-s − 8·61-s − 2·63-s + 7·64-s + 4·67-s + 3·73-s − 4·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.577·12-s + 1.10·13-s − 3/4·16-s + 0.917·19-s + 0.872·21-s + 2/5·25-s + 0.769·27-s + 0.377·28-s − 2.51·31-s − 1/6·36-s + 0.657·37-s − 1.28·39-s − 0.304·43-s + 0.866·48-s − 2/7·49-s − 0.554·52-s − 1.05·57-s − 1.02·61-s − 0.251·63-s + 7/8·64-s + 0.488·67-s + 0.351·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 657 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 657 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3119937432\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3119937432\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−14.84173356019134395733330207486, −14.08580238690987163680420223855, −13.41393326234914558128432992383, −12.86008664859908357365209131594, −12.24709854705253803421459273792, −11.23211640494482195728770394702, −11.11982146179022989852749019370, −10.11776565657729148299697331717, −9.279388737201940931926713800549, −8.726859407077604520825403564537, −7.47402283910884401831234750195, −6.54100615181400697590529104632, −5.81111174203705729174975290829, −4.91652301905022788110545307688, −3.56521670106926636818196449033,
3.56521670106926636818196449033, 4.91652301905022788110545307688, 5.81111174203705729174975290829, 6.54100615181400697590529104632, 7.47402283910884401831234750195, 8.726859407077604520825403564537, 9.279388737201940931926713800549, 10.11776565657729148299697331717, 11.11982146179022989852749019370, 11.23211640494482195728770394702, 12.24709854705253803421459273792, 12.86008664859908357365209131594, 13.41393326234914558128432992383, 14.08580238690987163680420223855, 14.84173356019134395733330207486