L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 16-s + 8·23-s − 25-s − 4·27-s − 8·29-s − 3·36-s + 20·43-s − 2·48-s + 5·49-s + 18·53-s − 12·61-s − 64-s − 16·69-s + 2·75-s + 12·79-s + 5·81-s + 16·87-s − 8·92-s + 100-s + 30·103-s + 4·107-s + 4·108-s − 16·113-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1/4·16-s + 1.66·23-s − 1/5·25-s − 0.769·27-s − 1.48·29-s − 1/2·36-s + 3.04·43-s − 0.288·48-s + 5/7·49-s + 2.47·53-s − 1.53·61-s − 1/8·64-s − 1.92·69-s + 0.230·75-s + 1.35·79-s + 5/9·81-s + 1.71·87-s − 0.834·92-s + 1/10·100-s + 2.95·103-s + 0.386·107-s + 0.384·108-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.373046105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.373046105\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.601210787736251452283567114581, −7.74376961124057779346326971116, −7.64893602029280037813170848950, −7.36133899707079316986979821576, −7.05417338786111630305243556601, −6.54785454165669105975724525716, −6.08890248327757243666290427977, −5.91226467228514597381191208541, −5.50409982882993187220297422579, −5.08416728227276067570101076998, −4.94550006468922623269185147684, −4.29256231449631530588458288255, −4.07542643680189639343869062064, −3.66066401784555330183746257264, −3.19507717556796983472041937912, −2.43506758899085231630649874501, −2.27518197208888701136112060493, −1.37005452738535426922838548994, −0.957216608346604294854305150230, −0.43778990589117934658639755139,
0.43778990589117934658639755139, 0.957216608346604294854305150230, 1.37005452738535426922838548994, 2.27518197208888701136112060493, 2.43506758899085231630649874501, 3.19507717556796983472041937912, 3.66066401784555330183746257264, 4.07542643680189639343869062064, 4.29256231449631530588458288255, 4.94550006468922623269185147684, 5.08416728227276067570101076998, 5.50409982882993187220297422579, 5.91226467228514597381191208541, 6.08890248327757243666290427977, 6.54785454165669105975724525716, 7.05417338786111630305243556601, 7.36133899707079316986979821576, 7.64893602029280037813170848950, 7.74376961124057779346326971116, 8.601210787736251452283567114581