| L(s) = 1 | + 4-s − 2·5-s − 7-s + 16-s − 4·17-s − 2·20-s − 7·25-s − 28-s + 2·35-s + 10·37-s + 18·41-s − 20·43-s − 12·47-s + 49-s + 28·59-s + 64-s − 16·67-s − 4·68-s + 12·79-s − 2·80-s + 8·83-s + 8·85-s + 18·89-s − 7·100-s + 28·101-s + 14·109-s − 112-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s − 0.377·7-s + 1/4·16-s − 0.970·17-s − 0.447·20-s − 7/5·25-s − 0.188·28-s + 0.338·35-s + 1.64·37-s + 2.81·41-s − 3.04·43-s − 1.75·47-s + 1/7·49-s + 3.64·59-s + 1/8·64-s − 1.95·67-s − 0.485·68-s + 1.35·79-s − 0.223·80-s + 0.878·83-s + 0.867·85-s + 1.90·89-s − 0.699·100-s + 2.78·101-s + 1.34·109-s − 0.0944·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000188 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000188 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−8.015044568391676191880321656496, −7.43396451799616319962951174764, −7.15629851020103523042213387688, −6.47784583599995832693494924500, −6.23227538486956081778076113568, −5.87450213364309522501266401702, −5.08207234913297047777508467306, −4.69069334189536458166730202416, −4.15202289907072344630083460584, −3.57313298834818934312909000630, −3.35511404251805033708538922757, −2.26551691900571842895890768283, −2.24049119020186278469176393363, −1.01936668222152272292011651270, 0,
1.01936668222152272292011651270, 2.24049119020186278469176393363, 2.26551691900571842895890768283, 3.35511404251805033708538922757, 3.57313298834818934312909000630, 4.15202289907072344630083460584, 4.69069334189536458166730202416, 5.08207234913297047777508467306, 5.87450213364309522501266401702, 6.23227538486956081778076113568, 6.47784583599995832693494924500, 7.15629851020103523042213387688, 7.43396451799616319962951174764, 8.015044568391676191880321656496
