L(s) = 1 | − 2·4-s + 7-s + 4·9-s + 4·11-s − 7·13-s − 7·17-s + 2·25-s − 2·28-s − 3·29-s − 8·36-s − 6·37-s − 4·43-s − 8·44-s − 7·47-s − 6·49-s + 14·52-s − 9·53-s − 7·59-s + 4·63-s + 8·64-s + 14·68-s + 4·77-s + 7·81-s + 7·89-s − 7·91-s + 7·97-s + 16·99-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s + 4/3·9-s + 1.20·11-s − 1.94·13-s − 1.69·17-s + 2/5·25-s − 0.377·28-s − 0.557·29-s − 4/3·36-s − 0.986·37-s − 0.609·43-s − 1.20·44-s − 1.02·47-s − 6/7·49-s + 1.94·52-s − 1.23·53-s − 0.911·59-s + 0.503·63-s + 64-s + 1.69·68-s + 0.455·77-s + 7/9·81-s + 0.741·89-s − 0.733·91-s + 0.710·97-s + 1.60·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 53 | $C_2$ | \( 1 + 9 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.123464599039727574412254112667, −8.878558082970661771699775238836, −8.136208381278717967532960881738, −7.62275587263343131877880006627, −7.12574955735069154540285203212, −6.61721018235880788371758471715, −6.38602252386875560510294136406, −5.14349971255565349607745065325, −4.92127655284803363739289981564, −4.45170359681593866525659157731, −4.07290051039862053056482493100, −3.26489009723204576040084360655, −2.17961200216846533417827059081, −1.56983437907093150983269767856, 0,
1.56983437907093150983269767856, 2.17961200216846533417827059081, 3.26489009723204576040084360655, 4.07290051039862053056482493100, 4.45170359681593866525659157731, 4.92127655284803363739289981564, 5.14349971255565349607745065325, 6.38602252386875560510294136406, 6.61721018235880788371758471715, 7.12574955735069154540285203212, 7.62275587263343131877880006627, 8.136208381278717967532960881738, 8.878558082970661771699775238836, 9.123464599039727574412254112667