| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·9-s − 8·13-s + 5·16-s + 6·17-s + 4·18-s + 4·19-s − 10·25-s + 16·26-s − 6·32-s − 12·34-s − 6·36-s − 8·38-s + 16·43-s − 24·47-s + 49-s + 20·50-s − 24·52-s + 12·53-s − 12·59-s + 7·64-s − 8·67-s + 18·68-s + 8·72-s + 12·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s − 2.21·13-s + 5/4·16-s + 1.45·17-s + 0.942·18-s + 0.917·19-s − 2·25-s + 3.13·26-s − 1.06·32-s − 2.05·34-s − 36-s − 1.29·38-s + 2.43·43-s − 3.50·47-s + 1/7·49-s + 2.82·50-s − 3.32·52-s + 1.64·53-s − 1.56·59-s + 7/8·64-s − 0.977·67-s + 2.18·68-s + 0.942·72-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.765547119459919407856461234632, −9.440338822267021662634471861951, −8.812642335551285956694248042850, −8.090068662745416752784621377032, −7.61488245805577994602083893658, −7.57571100088867902110310233811, −6.87015167727116618005628901277, −6.04516833074826431165424798565, −5.57928681742950427486583645839, −5.05791232432713970996781243183, −4.04379426959865020166490254054, −3.04116088637864497579983367990, −2.57969057241224021562840012921, −1.54182063036964932495656311078, 0,
1.54182063036964932495656311078, 2.57969057241224021562840012921, 3.04116088637864497579983367990, 4.04379426959865020166490254054, 5.05791232432713970996781243183, 5.57928681742950427486583645839, 6.04516833074826431165424798565, 6.87015167727116618005628901277, 7.57571100088867902110310233811, 7.61488245805577994602083893658, 8.090068662745416752784621377032, 8.812642335551285956694248042850, 9.440338822267021662634471861951, 9.765547119459919407856461234632
