| L(s) = 1 | − 4-s − 2·7-s + 4·9-s − 10·13-s − 3·16-s − 2·23-s − 6·25-s + 2·28-s − 4·29-s − 4·36-s − 3·49-s + 10·52-s + 12·53-s − 10·59-s − 8·63-s + 7·64-s + 14·67-s + 7·81-s − 20·83-s + 20·91-s + 2·92-s + 6·100-s + 10·103-s − 4·107-s + 6·112-s + 4·116-s − 40·117-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s + 4/3·9-s − 2.77·13-s − 3/4·16-s − 0.417·23-s − 6/5·25-s + 0.377·28-s − 0.742·29-s − 2/3·36-s − 3/7·49-s + 1.38·52-s + 1.64·53-s − 1.30·59-s − 1.00·63-s + 7/8·64-s + 1.71·67-s + 7/9·81-s − 2.19·83-s + 2.09·91-s + 0.208·92-s + 3/5·100-s + 0.985·103-s − 0.386·107-s + 0.566·112-s + 0.371·116-s − 3.69·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41209 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41209 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 29 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 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
−9.918103161223520397240654911344, −9.591238369246225749386482983285, −9.264853869188346104598010404109, −8.457197649962642253202077729058, −7.71240730416121449694136528785, −7.29155980463525000217122731746, −6.99392722222509128357175708398, −6.29458680310399886991101574008, −5.44779861172707230627218230153, −4.89946183719736252315133316436, −4.31367579764196367679844488173, −3.79916005521179829561097206512, −2.66866588116939972345119402788, −1.97638088946853147917783291957, 0,
1.97638088946853147917783291957, 2.66866588116939972345119402788, 3.79916005521179829561097206512, 4.31367579764196367679844488173, 4.89946183719736252315133316436, 5.44779861172707230627218230153, 6.29458680310399886991101574008, 6.99392722222509128357175708398, 7.29155980463525000217122731746, 7.71240730416121449694136528785, 8.457197649962642253202077729058, 9.264853869188346104598010404109, 9.591238369246225749386482983285, 9.918103161223520397240654911344
