| 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
| \(L(3)\) |
\(\approx\) |
\(2.236751583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.236751583\) |
| \(L(\frac{7}{2})\) |
|
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
−15.59964226317481342267609771744, −14.51953614573807688153170664752, −13.50198593976429974427420062163, −13.13838685224223409936876015023, −12.59492164439596816431039260967, −12.35743687166272342580647322623, −11.45177766655167248021121568451, −11.04988881648716627805627173981, −10.61744545586446670896712677784, −9.524240352591819466082394084054, −8.814950747871708736838933490252, −7.74602683652180958488176801987, −7.29948338626288867001525896968, −6.17941847040448502626987433539, −5.78738669456850938922649231359, −4.83201982646118943624290197710, −4.18272332541666713329523285204, −3.95185726408973716945506684858, −2.51225456828202737331967992047, −0.62543727073202147139343221438,
0.62543727073202147139343221438, 2.51225456828202737331967992047, 3.95185726408973716945506684858, 4.18272332541666713329523285204, 4.83201982646118943624290197710, 5.78738669456850938922649231359, 6.17941847040448502626987433539, 7.29948338626288867001525896968, 7.74602683652180958488176801987, 8.814950747871708736838933490252, 9.524240352591819466082394084054, 10.61744545586446670896712677784, 11.04988881648716627805627173981, 11.45177766655167248021121568451, 12.35743687166272342580647322623, 12.59492164439596816431039260967, 13.13838685224223409936876015023, 13.50198593976429974427420062163, 14.51953614573807688153170664752, 15.59964226317481342267609771744
