| L(s) = 1 | − 2·3-s − 4·5-s − 2·7-s + 2·9-s + 12·11-s − 2·13-s + 8·15-s + 2·17-s + 4·21-s + 10·23-s + 11·25-s − 6·27-s − 16·29-s − 24·33-s + 8·35-s − 10·37-s + 4·39-s − 12·41-s − 6·43-s − 8·45-s − 14·47-s + 2·49-s − 4·51-s + 2·53-s − 48·55-s − 4·63-s + 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 0.755·7-s + 2/3·9-s + 3.61·11-s − 0.554·13-s + 2.06·15-s + 0.485·17-s + 0.872·21-s + 2.08·23-s + 11/5·25-s − 1.15·27-s − 2.97·29-s − 4.17·33-s + 1.35·35-s − 1.64·37-s + 0.640·39-s − 1.87·41-s − 0.914·43-s − 1.19·45-s − 2.04·47-s + 2/7·49-s − 0.560·51-s + 0.274·53-s − 6.47·55-s − 0.503·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4427239784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4427239784\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.807735354794905581530592079613, −9.303216815984415969681090537131, −9.272072445653562646300314072668, −8.568888306870634378109142751078, −8.511109683962489576416988926912, −7.48038961464668760562434443057, −7.40452724078270674606199346892, −6.77957501910312377553814405254, −6.71416129302840148502117855105, −6.44056160427996727909260292567, −5.64444594093886384651758204454, −5.16737897252026642918622899044, −4.87166318245351595972247983051, −4.03753841264483963138599175312, −3.84986427928083685601883105354, −3.48688916936832019797727492340, −3.17707480947023233591466577047, −1.65171404125528601385392377384, −1.38527365421148303085833712471, −0.33044886942901098316539443010,
0.33044886942901098316539443010, 1.38527365421148303085833712471, 1.65171404125528601385392377384, 3.17707480947023233591466577047, 3.48688916936832019797727492340, 3.84986427928083685601883105354, 4.03753841264483963138599175312, 4.87166318245351595972247983051, 5.16737897252026642918622899044, 5.64444594093886384651758204454, 6.44056160427996727909260292567, 6.71416129302840148502117855105, 6.77957501910312377553814405254, 7.40452724078270674606199346892, 7.48038961464668760562434443057, 8.511109683962489576416988926912, 8.568888306870634378109142751078, 9.272072445653562646300314072668, 9.303216815984415969681090537131, 9.807735354794905581530592079613
