| L(s) = 1 | − 3-s − 2·7-s + 9-s − 12·13-s − 2·19-s + 2·21-s + 6·25-s − 27-s + 6·31-s + 14·37-s + 12·39-s + 8·43-s − 2·49-s + 2·57-s + 12·61-s − 2·63-s − 2·67-s − 4·73-s − 6·75-s + 81-s + 24·91-s − 6·93-s − 12·97-s − 24·103-s − 16·109-s − 14·111-s − 12·117-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 3.32·13-s − 0.458·19-s + 0.436·21-s + 6/5·25-s − 0.192·27-s + 1.07·31-s + 2.30·37-s + 1.92·39-s + 1.21·43-s − 2/7·49-s + 0.264·57-s + 1.53·61-s − 0.251·63-s − 0.244·67-s − 0.468·73-s − 0.692·75-s + 1/9·81-s + 2.51·91-s − 0.622·93-s − 1.21·97-s − 2.36·103-s − 1.53·109-s − 1.32·111-s − 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.031563942299162424041628843957, −7.63005705191795993983150082651, −7.21765743945199396325053287313, −6.69429743165646204191393746321, −6.60346364403284777452294576360, −5.74392577775493010279202459250, −5.47440673874821352532600321179, −4.73172476280743272790901464842, −4.57854700039207025851930473702, −4.06779230624302542715181771391, −3.02216390230599701572405821495, −2.62923448155770957999371438518, −2.25799243747642170485814685577, −0.934860212622062306194890180329, 0,
0.934860212622062306194890180329, 2.25799243747642170485814685577, 2.62923448155770957999371438518, 3.02216390230599701572405821495, 4.06779230624302542715181771391, 4.57854700039207025851930473702, 4.73172476280743272790901464842, 5.47440673874821352532600321179, 5.74392577775493010279202459250, 6.60346364403284777452294576360, 6.69429743165646204191393746321, 7.21765743945199396325053287313, 7.63005705191795993983150082651, 8.031563942299162424041628843957
