| L(s) = 1 | − 81·3-s + 126.·5-s − 5.93e3·7-s + 6.56e3·9-s − 4.23e3·11-s − 1.36e5·13-s − 1.02e4·15-s − 2.30e5·17-s − 4.50e5·19-s + 4.81e5·21-s − 1.33e6·23-s − 1.93e6·25-s − 5.31e5·27-s + 4.08e6·29-s + 4.71e6·31-s + 3.43e5·33-s − 7.53e5·35-s − 1.93e7·37-s + 1.10e7·39-s − 1.33e7·41-s − 4.48e6·43-s + 8.31e5·45-s + 1.27e7·47-s − 5.07e6·49-s + 1.86e7·51-s − 5.11e7·53-s − 5.37e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0907·5-s − 0.935·7-s + 0.333·9-s − 0.0872·11-s − 1.32·13-s − 0.0523·15-s − 0.668·17-s − 0.793·19-s + 0.539·21-s − 0.996·23-s − 0.991·25-s − 0.192·27-s + 1.07·29-s + 0.917·31-s + 0.0503·33-s − 0.0848·35-s − 1.69·37-s + 0.765·39-s − 0.738·41-s − 0.200·43-s + 0.0302·45-s + 0.382·47-s − 0.125·49-s + 0.386·51-s − 0.891·53-s − 0.00791·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2086957989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2086957989\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
| good | 5 | \( 1 - 126.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.93e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.23e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.36e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.30e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.50e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.33e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.08e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.71e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.93e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.33e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.48e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.27e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.11e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.93e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.72e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.24e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.69e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.14e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.35e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.48e8T + 7.60e17T^{2} \) |
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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
−10.05469483424188554483137838628, −9.037515126530044356390748176991, −7.893233507498670608765632544870, −6.79670333945907935456575095818, −6.19947333254074990508588305548, −5.05097220081053951928841175064, −4.12525058510336172980411296671, −2.84264302932600130754439240219, −1.80141486670372477889577922678, −0.18704348531330989235488134317,
0.18704348531330989235488134317, 1.80141486670372477889577922678, 2.84264302932600130754439240219, 4.12525058510336172980411296671, 5.05097220081053951928841175064, 6.19947333254074990508588305548, 6.79670333945907935456575095818, 7.893233507498670608765632544870, 9.037515126530044356390748176991, 10.05469483424188554483137838628
