| L(s) = 1 | + 2·2-s + 3·3-s + 2·4-s − 5-s + 6·6-s + 4·8-s + 3·9-s − 2·10-s − 11-s + 6·12-s − 6·13-s − 3·15-s + 8·16-s − 3·17-s + 6·18-s + 6·19-s − 2·20-s − 2·22-s + 4·23-s + 12·24-s − 12·26-s − 2·29-s − 6·30-s + 6·31-s + 8·32-s − 3·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.73·3-s + 4-s − 0.447·5-s + 2.44·6-s + 1.41·8-s + 9-s − 0.632·10-s − 0.301·11-s + 1.73·12-s − 1.66·13-s − 0.774·15-s + 2·16-s − 0.727·17-s + 1.41·18-s + 1.37·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s + 2.44·24-s − 2.35·26-s − 0.371·29-s − 1.09·30-s + 1.07·31-s + 1.41·32-s − 0.522·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.595102830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.595102830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
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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
−12.54972539759642104367415266528, −11.80518205282698511789905777449, −11.74377265472613533486315070680, −11.07682448847542967839942019117, −10.28256014711089096364837300458, −9.914595807126040473686747426724, −9.585831038598588300180420944100, −8.573827798231678959576283494924, −8.544040517581890388977522647801, −7.84274556150422326450072855576, −7.21177849114347979778405506475, −7.20004770810884308462970841119, −6.27741063323307409923161103976, −5.13242414711524431103582956222, −5.06096741857964967920932525140, −4.48396450632544360451467890435, −3.59394641062905211757012566824, −3.20350596770247642540931548808, −2.64988853482532921135503103236, −1.80439682841877508434819750501,
1.80439682841877508434819750501, 2.64988853482532921135503103236, 3.20350596770247642540931548808, 3.59394641062905211757012566824, 4.48396450632544360451467890435, 5.06096741857964967920932525140, 5.13242414711524431103582956222, 6.27741063323307409923161103976, 7.20004770810884308462970841119, 7.21177849114347979778405506475, 7.84274556150422326450072855576, 8.544040517581890388977522647801, 8.573827798231678959576283494924, 9.585831038598588300180420944100, 9.914595807126040473686747426724, 10.28256014711089096364837300458, 11.07682448847542967839942019117, 11.74377265472613533486315070680, 11.80518205282698511789905777449, 12.54972539759642104367415266528
