| L(s) = 1 | + 303.·5-s + 260·7-s + 6.07e3·11-s − 6.89e3·13-s + 2.36e4·17-s − 3.31e4·19-s + 3.15e4·23-s + 1.40e4·25-s + 1.38e5·29-s + 1.50e3·31-s + 7.89e4·35-s + 3.80e5·37-s + 8.80e4·41-s − 7.64e3·43-s + 5.65e5·47-s − 7.55e5·49-s − 1.03e6·53-s + 1.84e6·55-s + 2.70e6·59-s + 9.88e5·61-s − 2.09e6·65-s − 3.85e6·67-s + 4.22e6·71-s − 2.00e6·73-s + 1.57e6·77-s + 2.69e6·79-s − 2.71e6·83-s + ⋯ |
| L(s) = 1 | + 1.08·5-s + 0.286·7-s + 1.37·11-s − 0.869·13-s + 1.16·17-s − 1.10·19-s + 0.541·23-s + 0.179·25-s + 1.05·29-s + 0.00909·31-s + 0.311·35-s + 1.23·37-s + 0.199·41-s − 0.0146·43-s + 0.795·47-s − 0.917·49-s − 0.950·53-s + 1.49·55-s + 1.71·59-s + 0.557·61-s − 0.944·65-s − 1.56·67-s + 1.40·71-s − 0.603·73-s + 0.394·77-s + 0.616·79-s − 0.520·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.543601244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.543601244\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 303.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 260T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.89e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.50e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.80e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.64e3T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.65e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.70e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.88e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.85e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.00e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.69e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.74e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.29e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.649308985651729339032341149033, −8.884578551916667588837164570257, −7.84973361605040593940565474150, −6.73521624351883338122584758387, −6.05663812758533629441239668206, −5.05843793895807687518485294376, −4.06758413421051476783688383672, −2.75374233228751384931701464366, −1.76081158777386765796330344439, −0.844199808316436994349770644860,
0.844199808316436994349770644860, 1.76081158777386765796330344439, 2.75374233228751384931701464366, 4.06758413421051476783688383672, 5.05843793895807687518485294376, 6.05663812758533629441239668206, 6.73521624351883338122584758387, 7.84973361605040593940565474150, 8.884578551916667588837164570257, 9.649308985651729339032341149033
