| L(s) = 1 | + 2·3-s + 3·9-s + 4·13-s − 4·16-s + 6·17-s + 18·23-s + 4·27-s + 8·39-s − 12·43-s − 8·48-s + 5·49-s + 12·51-s + 18·53-s − 6·61-s + 36·69-s − 30·79-s + 5·81-s + 24·101-s − 12·103-s + 6·107-s − 12·113-s + 12·117-s − 3·121-s + 127-s − 24·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.10·13-s − 16-s + 1.45·17-s + 3.75·23-s + 0.769·27-s + 1.28·39-s − 1.82·43-s − 1.15·48-s + 5/7·49-s + 1.68·51-s + 2.47·53-s − 0.768·61-s + 4.33·69-s − 3.37·79-s + 5/9·81-s + 2.38·101-s − 1.18·103-s + 0.580·107-s − 1.12·113-s + 1.10·117-s − 0.272·121-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.091687807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.091687807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 177 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−10.05652678864284505373082740345, −9.844555487214005168548192408041, −9.128627404364523030585087529706, −8.812953021194253297494173466794, −8.810674391743879450360032204337, −8.312244210782529717156073509378, −7.61640013576733680858872446314, −7.35482813972291040714348341491, −6.89509460255365663843260279291, −6.63409788865509282531324629436, −5.92567536080961044582050201637, −5.23830175196200740721819086433, −5.10805099801321030107831096760, −4.33741440911906687938561247947, −3.89436181809710876868877923771, −3.12085608877598939943627032573, −3.11393859532465136887295305191, −2.40467050773086630571974223318, −1.43834458980090649132042351972, −1.02352282644378969284708846151,
1.02352282644378969284708846151, 1.43834458980090649132042351972, 2.40467050773086630571974223318, 3.11393859532465136887295305191, 3.12085608877598939943627032573, 3.89436181809710876868877923771, 4.33741440911906687938561247947, 5.10805099801321030107831096760, 5.23830175196200740721819086433, 5.92567536080961044582050201637, 6.63409788865509282531324629436, 6.89509460255365663843260279291, 7.35482813972291040714348341491, 7.61640013576733680858872446314, 8.312244210782529717156073509378, 8.810674391743879450360032204337, 8.812953021194253297494173466794, 9.128627404364523030585087529706, 9.844555487214005168548192408041, 10.05652678864284505373082740345
