| L(s) = 1 | + 4-s − 2·7-s − 2·9-s − 6·11-s − 2·13-s + 16-s − 4·25-s − 2·28-s − 2·31-s − 2·36-s − 6·44-s − 2·49-s − 2·52-s − 6·53-s − 8·61-s + 4·63-s + 64-s + 12·77-s − 5·81-s − 6·83-s + 4·91-s + 16·97-s + 12·99-s − 4·100-s − 2·109-s − 2·112-s − 6·113-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s − 2/3·9-s − 1.80·11-s − 0.554·13-s + 1/4·16-s − 4/5·25-s − 0.377·28-s − 0.359·31-s − 1/3·36-s − 0.904·44-s − 2/7·49-s − 0.277·52-s − 0.824·53-s − 1.02·61-s + 0.503·63-s + 1/8·64-s + 1.36·77-s − 5/9·81-s − 0.658·83-s + 0.419·91-s + 1.62·97-s + 1.20·99-s − 2/5·100-s − 0.191·109-s − 0.188·112-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51076 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51076 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 113 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 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
−9.862712142604085049209623611431, −9.452509679277657260960775296764, −8.772547206064145308518338707094, −8.212130933668915115609201169937, −7.60782730954084959812831816858, −7.44286755429265695245563803410, −6.55452630286274930031514542996, −6.11575715390277190493892874669, −5.46271450554253433650790454625, −5.07358718995726083799617436806, −4.19059837751793309437580208697, −3.19573070181924713266714440631, −2.81686781992177357289070727538, −2.00585867734746745170902909992, 0,
2.00585867734746745170902909992, 2.81686781992177357289070727538, 3.19573070181924713266714440631, 4.19059837751793309437580208697, 5.07358718995726083799617436806, 5.46271450554253433650790454625, 6.11575715390277190493892874669, 6.55452630286274930031514542996, 7.44286755429265695245563803410, 7.60782730954084959812831816858, 8.212130933668915115609201169937, 8.772547206064145308518338707094, 9.452509679277657260960775296764, 9.862712142604085049209623611431
