| 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
−9.601480167620329050572706184489, −7.87116874885778917827887965347, −7.48594399595209335177942333198, −6.26764441680948354586921979184, −5.33981553902281266196181237376, −4.94030840822155003036487027634, −3.69808575328628708085332965062, −2.82953641587468040425454832270, −1.95714595853659062245675920535, 0,
1.95714595853659062245675920535, 2.82953641587468040425454832270, 3.69808575328628708085332965062, 4.94030840822155003036487027634, 5.33981553902281266196181237376, 6.26764441680948354586921979184, 7.48594399595209335177942333198, 7.87116874885778917827887965347, 9.601480167620329050572706184489
