| L(s) = 1 | − 10·2-s + 68·4-s − 6·7-s − 360·8-s + 685·11-s − 685·13-s + 60·14-s + 1.42e3·16-s − 1.04e3·17-s + 1.10e3·19-s − 6.85e3·22-s + 3.85e3·23-s + 6.85e3·26-s − 408·28-s − 1.31e3·29-s − 9.90e3·31-s − 2.72e3·32-s + 1.04e4·34-s + 6.82e3·37-s − 1.10e4·38-s + 4.52e3·41-s − 9.09e3·43-s + 4.65e4·44-s − 3.85e4·46-s − 2.09e3·47-s − 1.67e4·49-s − 4.65e4·52-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s − 0.0462·7-s − 1.98·8-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 − 3.01·22-s + 1.51·23-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 + 0.819·37-s − 1.24·38-s + 0.419·41-s − 0.750·43-s + 3.62·44-s − 2.68·46-s − 0.138·47-s − 0.997·49-s − 2.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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
−9.252485285320326147151048570441, −8.789298265778406321856641145251, −7.52677449838576785469865916867, −7.04812779237676035728560350731, −6.19277198916445977514723803994, −4.73879480212551970237876321244, −3.28707869336835419445608309687, −2.03155713660153038673659558886, −1.12490980589120367866702008003, 0,
1.12490980589120367866702008003, 2.03155713660153038673659558886, 3.28707869336835419445608309687, 4.73879480212551970237876321244, 6.19277198916445977514723803994, 7.04812779237676035728560350731, 7.52677449838576785469865916867, 8.789298265778406321856641145251, 9.252485285320326147151048570441
