| L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 2·13-s + 6·17-s + 4·21-s − 6·23-s + 4·27-s − 6·29-s + 4·31-s + 2·37-s − 4·39-s − 6·41-s + 10·43-s + 6·47-s − 3·49-s − 12·51-s − 6·53-s − 12·59-s + 2·61-s − 2·63-s + 2·67-s + 12·69-s + 12·71-s − 2·73-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 0.872·21-s − 1.25·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.251·63-s + 0.244·67-s + 1.44·69-s + 1.42·71-s − 0.234·73-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.49774404899198, −14.48179522890825, −14.16856209370504, −13.61245115783339, −12.83514181328348, −12.51921566817818, −12.02090388977209, −11.50329876311137, −11.04107095235103, −10.33837861704804, −10.06870523182061, −9.398866521945697, −8.862669328017785, −7.946805593009705, −7.701563645291269, −6.806063579786096, −6.272196260757473, −5.849803824146301, −5.453873457762236, −4.690208479323645, −3.937677558970108, −3.375263162050059, −2.650834581082384, −1.622442630699115, −0.8215993205563163, 0,
0.8215993205563163, 1.622442630699115, 2.650834581082384, 3.375263162050059, 3.937677558970108, 4.690208479323645, 5.453873457762236, 5.849803824146301, 6.272196260757473, 6.806063579786096, 7.701563645291269, 7.946805593009705, 8.862669328017785, 9.398866521945697, 10.06870523182061, 10.33837861704804, 11.04107095235103, 11.50329876311137, 12.02090388977209, 12.51921566817818, 12.83514181328348, 13.61245115783339, 14.16856209370504, 14.48179522890825, 15.49774404899198
