L(s) = 1 | + 3-s + 4-s − 2·9-s + 12-s − 3·16-s + 3·17-s − 12·23-s − 2·25-s − 5·27-s − 6·29-s − 2·36-s + 2·43-s − 3·48-s + 4·49-s + 3·51-s − 9·53-s − 2·61-s − 7·64-s + 3·68-s − 12·69-s − 2·75-s − 17·79-s + 81-s − 6·87-s − 12·92-s − 2·100-s + 24·101-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/2·4-s − 2/3·9-s + 0.288·12-s − 3/4·16-s + 0.727·17-s − 2.50·23-s − 2/5·25-s − 0.962·27-s − 1.11·29-s − 1/3·36-s + 0.304·43-s − 0.433·48-s + 4/7·49-s + 0.420·51-s − 1.23·53-s − 0.256·61-s − 7/8·64-s + 0.363·68-s − 1.44·69-s − 0.230·75-s − 1.91·79-s + 1/9·81-s − 0.643·87-s − 1.25·92-s − 1/5·100-s + 2.38·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | | \( 1 \) |
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 137 T^{2} + p^{2} T^{4} \) |
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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
−8.605765727387772171671177448844, −8.203212520718643006382035873563, −7.73854214297040726227504963296, −7.48297150439161013552716091134, −6.82220946742860293131329804574, −6.21667101523989003188493344219, −5.79791006136415345619960604182, −5.51025949816500121086584961598, −4.59107975065269835474260216476, −4.05048738683211302669525375001, −3.52616879418838437365718740972, −2.86645069959728951184932408402, −2.19671714828714623524096750224, −1.69853921679169847159914749162, 0,
1.69853921679169847159914749162, 2.19671714828714623524096750224, 2.86645069959728951184932408402, 3.52616879418838437365718740972, 4.05048738683211302669525375001, 4.59107975065269835474260216476, 5.51025949816500121086584961598, 5.79791006136415345619960604182, 6.21667101523989003188493344219, 6.82220946742860293131329804574, 7.48297150439161013552716091134, 7.73854214297040726227504963296, 8.203212520718643006382035873563, 8.605765727387772171671177448844