| L(s) = 1 | + 24·3-s − 25·5-s + 172·7-s + 333·9-s + 132·11-s + 946·13-s − 600·15-s − 222·17-s + 500·19-s + 4.12e3·21-s − 3.56e3·23-s + 625·25-s + 2.16e3·27-s − 2.19e3·29-s − 2.31e3·31-s + 3.16e3·33-s − 4.30e3·35-s + 1.12e4·37-s + 2.27e4·39-s + 1.24e3·41-s + 2.06e4·43-s − 8.32e3·45-s − 6.58e3·47-s + 1.27e4·49-s − 5.32e3·51-s + 2.10e4·53-s − 3.30e3·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 0.447·5-s + 1.32·7-s + 1.37·9-s + 0.328·11-s + 1.55·13-s − 0.688·15-s − 0.186·17-s + 0.317·19-s + 2.04·21-s − 1.40·23-s + 1/5·25-s + 0.570·27-s − 0.483·29-s − 0.432·31-s + 0.506·33-s − 0.593·35-s + 1.35·37-s + 2.39·39-s + 0.115·41-s + 1.70·43-s − 0.612·45-s − 0.435·47-s + 0.760·49-s − 0.286·51-s + 1.03·53-s − 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.390382224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.390382224\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 172 T + p^{5} T^{2} \) |
| 11 | \( 1 - 12 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 946 T + p^{5} T^{2} \) |
| 17 | \( 1 + 222 T + p^{5} T^{2} \) |
| 19 | \( 1 - 500 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3564 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2190 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2312 T + p^{5} T^{2} \) |
| 37 | \( 1 - 11242 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1242 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20624 T + p^{5} T^{2} \) |
| 47 | \( 1 + 6588 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21066 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7980 T + p^{5} T^{2} \) |
| 61 | \( 1 + 16622 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1808 T + p^{5} T^{2} \) |
| 71 | \( 1 - 24528 T + p^{5} T^{2} \) |
| 73 | \( 1 - 20474 T + p^{5} T^{2} \) |
| 79 | \( 1 - 46240 T + p^{5} T^{2} \) |
| 83 | \( 1 + 51576 T + p^{5} T^{2} \) |
| 89 | \( 1 + 110310 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78382 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90924396603737368037015595992, −9.588250647013204593507774614761, −8.658164998803719981906157814389, −8.129382058323387582760828079527, −7.40889470197565002955257992441, −5.89347488017721842263381406763, −4.31204954189937497484227371871, −3.65229857340425704103033894094, −2.26352991697923034203993830060, −1.22474548185061226711237696972,
1.22474548185061226711237696972, 2.26352991697923034203993830060, 3.65229857340425704103033894094, 4.31204954189937497484227371871, 5.89347488017721842263381406763, 7.40889470197565002955257992441, 8.129382058323387582760828079527, 8.658164998803719981906157814389, 9.588250647013204593507774614761, 10.90924396603737368037015595992
