| L(s) = 1 | + 18·3-s − 5·4-s − 100·7-s + 243·9-s − 90·12-s − 231·16-s − 1.80e3·21-s + 1.10e3·25-s + 2.91e3·27-s + 500·28-s − 1.21e3·36-s + 2.73e3·37-s − 4.15e3·48-s + 2.69e3·49-s − 2.43e4·63-s + 2.43e3·64-s − 1.78e4·67-s + 2.10e4·73-s + 1.98e4·75-s + 3.28e4·81-s + 9.00e3·84-s − 5.51e3·100-s − 1.45e4·108-s + 4.92e4·111-s + 2.31e4·112-s + 2.92e4·121-s + 127-s + ⋯ |
| L(s) = 1 | + 2·3-s − 0.312·4-s − 2.04·7-s + 3·9-s − 5/8·12-s − 0.902·16-s − 4.08·21-s + 1.76·25-s + 4·27-s + 0.637·28-s − 0.937·36-s + 2·37-s − 1.80·48-s + 1.12·49-s − 6.12·63-s + 0.594·64-s − 3.97·67-s + 3.94·73-s + 3.52·75-s + 5·81-s + 1.27·84-s − 0.550·100-s − 5/4·108-s + 4·111-s + 1.84·112-s + 2·121-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.585278691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.585278691\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 5 T^{2} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 1102 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 50 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 88750 T^{2} + p^{8} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 229010 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22030 T^{2} + p^{8} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 21208270 T^{2} + p^{8} T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8930 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10510 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 124695790 T^{2} + p^{8} T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−13.25390567267843622015481328771, −12.72483933051850035098511194869, −12.71620275413209744193476107162, −11.83005899053500747933060055095, −10.77344171077296525388672473513, −10.39719566921935704664796605002, −9.598802653189385177884112965806, −9.372803696263154436030096273010, −9.142465159719417450514039114099, −8.369792733854610376694142748604, −7.88085995415508905571778186058, −6.94948497518801022544382182135, −6.82383830362466650434022846923, −6.04329458083799886707156826566, −4.69195078256486575245008588012, −4.21824184342969040939783965400, −3.12656102701201310752712220570, −3.12041954328504683873998370732, −2.13534813519323308049074592402, −0.75449078467963060012573272581,
0.75449078467963060012573272581, 2.13534813519323308049074592402, 3.12041954328504683873998370732, 3.12656102701201310752712220570, 4.21824184342969040939783965400, 4.69195078256486575245008588012, 6.04329458083799886707156826566, 6.82383830362466650434022846923, 6.94948497518801022544382182135, 7.88085995415508905571778186058, 8.369792733854610376694142748604, 9.142465159719417450514039114099, 9.372803696263154436030096273010, 9.598802653189385177884112965806, 10.39719566921935704664796605002, 10.77344171077296525388672473513, 11.83005899053500747933060055095, 12.71620275413209744193476107162, 12.72483933051850035098511194869, 13.25390567267843622015481328771
