| L(s) = 1 | − 4·3-s − 2·4-s + 6·9-s + 6·11-s + 8·12-s + 4·16-s − 6·17-s + 2·19-s − 25-s + 4·27-s − 24·33-s − 12·36-s − 12·41-s − 2·43-s − 12·44-s − 16·48-s − 13·49-s + 24·51-s − 8·57-s − 12·59-s − 8·64-s − 8·67-s + 12·68-s − 14·73-s + 4·75-s − 4·76-s − 37·81-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 4-s + 2·9-s + 1.80·11-s + 2.30·12-s + 16-s − 1.45·17-s + 0.458·19-s − 1/5·25-s + 0.769·27-s − 4.17·33-s − 2·36-s − 1.87·41-s − 0.304·43-s − 1.80·44-s − 2.30·48-s − 1.85·49-s + 3.36·51-s − 1.05·57-s − 1.56·59-s − 64-s − 0.977·67-s + 1.45·68-s − 1.63·73-s + 0.461·75-s − 0.458·76-s − 4.11·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.49533789550642470301568900791, −10.24320445923116880350231460909, −9.199158339911279140575946730378, −9.180168549070062598241568188033, −8.462914393396165009068265832341, −7.65894969899749366452649043261, −6.68601933176938591059893570713, −6.39084306102520193435942181609, −6.07700564140158959836320975863, −5.03912355415234199771433602492, −4.96903618429312636799150610379, −4.19306407725294924458030580762, −3.33866387034377157662854676826, −1.39430917521719853589747801849, 0,
1.39430917521719853589747801849, 3.33866387034377157662854676826, 4.19306407725294924458030580762, 4.96903618429312636799150610379, 5.03912355415234199771433602492, 6.07700564140158959836320975863, 6.39084306102520193435942181609, 6.68601933176938591059893570713, 7.65894969899749366452649043261, 8.462914393396165009068265832341, 9.180168549070062598241568188033, 9.199158339911279140575946730378, 10.24320445923116880350231460909, 10.49533789550642470301568900791
