| L(s) = 1 | − 12·3-s + 36·5-s + 15·9-s − 18·11-s − 432·15-s + 720·17-s + 156·19-s − 738·23-s + 239·25-s + 396·27-s − 1.69e3·29-s − 2.01e3·31-s + 216·33-s − 2.38e3·37-s − 5.02e3·43-s + 540·45-s − 5.90e3·47-s − 8.64e3·51-s + 270·53-s − 648·55-s − 1.87e3·57-s + 5.43e3·59-s + 1.12e4·61-s − 2.45e3·67-s + 8.85e3·69-s − 6.30e3·71-s − 408·73-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 1.43·5-s + 5/27·9-s − 0.148·11-s − 1.91·15-s + 2.49·17-s + 0.432·19-s − 1.39·23-s + 0.382·25-s + 0.543·27-s − 2.01·29-s − 2.09·31-s + 0.198·33-s − 1.74·37-s − 2.71·43-s + 4/15·45-s − 2.67·47-s − 3.32·51-s + 0.0961·53-s − 0.214·55-s − 0.576·57-s + 1.56·59-s + 3.02·61-s − 0.545·67-s + 1.86·69-s − 1.24·71-s − 0.0765·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8410758688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8410758688\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 p T + 43 p T^{2} + 4 p^{5} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 36 T + 1057 T^{2} - 36 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 18 T - 14317 T^{2} + 18 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 39794 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 720 T + 256321 T^{2} - 720 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 156 T + 138433 T^{2} - 156 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 738 T + 264803 T^{2} + 738 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 846 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2016 T + 2278273 T^{2} + 2016 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2386 T + 3818835 T^{2} + 2386 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1634 p^{2} T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2510 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 5904 T + 16498753 T^{2} + 5904 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 270 T - 7817581 T^{2} - 270 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5436 T + 21967393 T^{2} - 5436 p^{4} T^{3} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11244 T + 55988353 T^{2} - 11244 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2450 T - 14148621 T^{2} + 2450 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3150 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 408 T + 28453729 T^{2} + 408 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3982 T - 23093757 T^{2} - 3982 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 69825650 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 13176 T + 120611233 T^{2} + 13176 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18761474 T^{2} + p^{8} T^{4} \) |
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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
−11.88366263547065968016933145115, −11.49568098572837921004098862982, −11.37216684385328872316458219242, −10.42684142448880565150605161849, −10.13795634156768687575203763349, −9.679126069723734556737984052571, −9.520671917401818397499633103544, −8.375238864942480962976237920729, −8.200433030295041251025511424002, −7.17088915339566772087372053512, −6.96001115844032002459025216658, −5.86757947647738760262608666939, −5.78675766438376534405104777774, −5.40612305329253897162086842127, −5.04435393657638649830475050314, −3.58811204779247745131890719939, −3.39512219007401818476142185935, −1.77577272859877898899702164867, −1.77358126583850566450466793460, −0.34515227332931472146103882234,
0.34515227332931472146103882234, 1.77358126583850566450466793460, 1.77577272859877898899702164867, 3.39512219007401818476142185935, 3.58811204779247745131890719939, 5.04435393657638649830475050314, 5.40612305329253897162086842127, 5.78675766438376534405104777774, 5.86757947647738760262608666939, 6.96001115844032002459025216658, 7.17088915339566772087372053512, 8.200433030295041251025511424002, 8.375238864942480962976237920729, 9.520671917401818397499633103544, 9.679126069723734556737984052571, 10.13795634156768687575203763349, 10.42684142448880565150605161849, 11.37216684385328872316458219242, 11.49568098572837921004098862982, 11.88366263547065968016933145115
