| L(s) = 1 | + 10·2-s + 14·3-s + 32·4-s + 56·5-s + 140·6-s − 40·8-s + 243·9-s + 560·10-s − 232·11-s + 448·12-s − 280·13-s + 784·15-s − 400·16-s + 1.72e3·17-s + 2.43e3·18-s + 98·19-s + 1.79e3·20-s − 2.32e3·22-s − 1.82e3·23-s − 560·24-s + 3.12e3·25-s − 2.80e3·26-s + 7.46e3·27-s + 6.83e3·29-s + 7.84e3·30-s + 7.64e3·31-s − 1.28e3·32-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 0.898·3-s + 4-s + 1.00·5-s + 1.58·6-s − 0.220·8-s + 9-s + 1.77·10-s − 0.578·11-s + 0.898·12-s − 0.459·13-s + 0.899·15-s − 0.390·16-s + 1.44·17-s + 1.76·18-s + 0.0622·19-s + 1.00·20-s − 1.02·22-s − 0.718·23-s − 0.198·24-s + 25-s − 0.812·26-s + 1.96·27-s + 1.50·29-s + 1.59·30-s + 1.42·31-s − 0.220·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.619035145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.619035145\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 5 p T + 17 p^{2} T^{2} - 5 p^{6} T^{3} + p^{10} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 14 T - 47 T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 56 T + 11 T^{2} - 56 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 232 T - 107227 T^{2} + 232 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 140 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1722 T + 1545427 T^{2} - 1722 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 98 T - 2466495 T^{2} - 98 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 1824 T - 3109367 T^{2} + 1824 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3418 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 7644 T + 29801585 T^{2} - 7644 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10398 T + 38774447 T^{2} - 10398 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 17962 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10880 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 9324 T - 142408031 T^{2} + 9324 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2262 T - 413078849 T^{2} + 2262 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2730 T - 707471399 T^{2} - 2730 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 25648 T - 186776397 T^{2} + 25648 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 48404 T + 992822109 T^{2} - 48404 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 58560 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 68082 T + 2562087131 T^{2} + 68082 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 31784 T - 2066833743 T^{2} + 31784 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 20538 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 50582 T - 3025520725 T^{2} - 50582 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 58506 T + p^{5} T^{2} )^{2} \) |
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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
−14.75116147476266323717186872013, −13.99019392852674182704569846040, −13.92782457534689697449612265746, −13.32225650734340726005640742125, −12.89535479055433536770676316372, −12.12003271570254587440069436182, −12.06421732189330948197998792237, −10.56829277251978858596398777022, −10.03059082367094244500559817504, −9.794226102250046364139637746384, −8.679755931679301164602667279628, −8.160881381965937121305945186967, −7.24716298457122185305366346917, −6.36481384511120754955120062540, −5.64061222361562283675576668981, −4.82713650170634663430257721032, −4.41601529893608581781144357152, −3.17513189780478372335215489900, −2.72501862716356878672501023234, −1.31635470525191252938833229704,
1.31635470525191252938833229704, 2.72501862716356878672501023234, 3.17513189780478372335215489900, 4.41601529893608581781144357152, 4.82713650170634663430257721032, 5.64061222361562283675576668981, 6.36481384511120754955120062540, 7.24716298457122185305366346917, 8.160881381965937121305945186967, 8.679755931679301164602667279628, 9.794226102250046364139637746384, 10.03059082367094244500559817504, 10.56829277251978858596398777022, 12.06421732189330948197998792237, 12.12003271570254587440069436182, 12.89535479055433536770676316372, 13.32225650734340726005640742125, 13.92782457534689697449612265746, 13.99019392852674182704569846040, 14.75116147476266323717186872013
