| L(s) = 1 | + 3·7-s − 2·11-s − 3·13-s − 6·17-s − 7·19-s − 6·23-s + 2·29-s − 5·31-s + 10·37-s − 12·41-s + 3·43-s + 10·47-s + 2·49-s + 6·59-s − 13·61-s + 7·67-s + 4·71-s − 6·73-s − 6·77-s − 8·79-s + 6·83-s − 16·89-s − 9·91-s − 7·97-s − 12·103-s + 16·107-s + 9·109-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 0.603·11-s − 0.832·13-s − 1.45·17-s − 1.60·19-s − 1.25·23-s + 0.371·29-s − 0.898·31-s + 1.64·37-s − 1.87·41-s + 0.457·43-s + 1.45·47-s + 2/7·49-s + 0.781·59-s − 1.66·61-s + 0.855·67-s + 0.474·71-s − 0.702·73-s − 0.683·77-s − 0.900·79-s + 0.658·83-s − 1.69·89-s − 0.943·91-s − 0.710·97-s − 1.18·103-s + 1.54·107-s + 0.862·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.687317716187599920863113173278, −8.185950120514054282770297981763, −7.39251670550150742087066701310, −6.51300867975066392569978306528, −5.58239379032457876412961380567, −4.62488497702715547946506468996, −4.15167539162411011404326170827, −2.52407450006420425610946316785, −1.88730569082121758267088569211, 0,
1.88730569082121758267088569211, 2.52407450006420425610946316785, 4.15167539162411011404326170827, 4.62488497702715547946506468996, 5.58239379032457876412961380567, 6.51300867975066392569978306528, 7.39251670550150742087066701310, 8.185950120514054282770297981763, 8.687317716187599920863113173278
