| L(s) = 1 | − 10·2-s + 68·4-s + 25·5-s + 6·7-s − 360·8-s − 250·10-s − 685·11-s + 685·13-s − 60·14-s + 1.42e3·16-s − 1.04e3·17-s + 1.10e3·19-s + 1.70e3·20-s + 6.85e3·22-s + 3.85e3·23-s + 625·25-s − 6.85e3·26-s + 408·28-s + 1.31e3·29-s − 9.90e3·31-s − 2.72e3·32-s + 1.04e4·34-s + 150·35-s − 6.82e3·37-s − 1.10e4·38-s − 9.00e3·40-s − 4.52e3·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s + 0.447·5-s + 0.0462·7-s − 1.98·8-s − 0.790·10-s − 1.70·11-s + 1.12·13-s − 0.0818·14-s + 1.39·16-s − 0.876·17-s + 0.704·19-s + 0.950·20-s + 3.01·22-s + 1.51·23-s + 1/5·25-s − 1.98·26-s + 0.0983·28-s + 0.290·29-s − 1.85·31-s − 0.469·32-s + 1.55·34-s + 0.0206·35-s − 0.819·37-s − 1.24·38-s − 0.889·40-s − 0.419·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 6 T + p^{5} T^{2} \) |
| 11 | \( 1 + 685 T + p^{5} T^{2} \) |
| 13 | \( 1 - 685 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1045 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1108 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3855 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1315 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9909 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6826 T + p^{5} T^{2} \) |
| 41 | \( 1 + 4520 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9097 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2095 T + p^{5} T^{2} \) |
| 53 | \( 1 + 10060 T + p^{5} T^{2} \) |
| 59 | \( 1 + 24820 T + p^{5} T^{2} \) |
| 61 | \( 1 + 46286 T + p^{5} T^{2} \) |
| 67 | \( 1 - 13860 T + p^{5} T^{2} \) |
| 71 | \( 1 - 75580 T + p^{5} T^{2} \) |
| 73 | \( 1 + 32738 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74877 T + p^{5} T^{2} \) |
| 83 | \( 1 + 93930 T + p^{5} T^{2} \) |
| 89 | \( 1 + 123540 T + p^{5} T^{2} \) |
| 97 | \( 1 + 85966 T + p^{5} T^{2} \) |
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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
−10.95991465584103411565172904708, −10.85420896671507387230405331735, −9.552745709035985184415861790251, −8.737422121807437761046319793900, −7.76890543661077048326596364671, −6.73806448346787250521742811048, −5.34607154789907389230600586702, −2.87990119759932974892112565879, −1.51324385168659524044988185275, 0,
1.51324385168659524044988185275, 2.87990119759932974892112565879, 5.34607154789907389230600586702, 6.73806448346787250521742811048, 7.76890543661077048326596364671, 8.737422121807437761046319793900, 9.552745709035985184415861790251, 10.85420896671507387230405331735, 10.95991465584103411565172904708
