L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 9-s − 3·11-s + 2·13-s − 2·15-s + 6·17-s + 8·21-s + 25-s + 4·27-s − 3·29-s + 7·31-s + 6·33-s − 4·35-s + 8·37-s − 4·39-s + 6·41-s + 4·43-s + 45-s + 6·47-s + 9·49-s − 12·51-s − 6·53-s − 3·55-s − 15·59-s − 5·61-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 1.74·21-s + 1/5·25-s + 0.769·27-s − 0.557·29-s + 1.25·31-s + 1.04·33-s − 0.676·35-s + 1.31·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.875·47-s + 9/7·49-s − 1.68·51-s − 0.824·53-s − 0.404·55-s − 1.95·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.85534190091318, −13.18532536097134, −12.78446592408340, −12.43118986046997, −12.05854071688336, −11.33278405448824, −10.93077415358047, −10.39696585749142, −10.09352098641861, −9.482931264667413, −9.225052537702406, −8.391648250908707, −7.755629132673802, −7.402042100084228, −6.567577514471714, −6.164995758009329, −5.913272356363393, −5.452235758866822, −4.820269811242422, −4.183458565450543, −3.413784288976459, −2.893465695160747, −2.466969055561507, −1.285085640481848, −0.7544475868841402, 0,
0.7544475868841402, 1.285085640481848, 2.466969055561507, 2.893465695160747, 3.413784288976459, 4.183458565450543, 4.820269811242422, 5.452235758866822, 5.913272356363393, 6.164995758009329, 6.567577514471714, 7.402042100084228, 7.755629132673802, 8.391648250908707, 9.225052537702406, 9.482931264667413, 10.09352098641861, 10.39696585749142, 10.93077415358047, 11.33278405448824, 12.05854071688336, 12.43118986046997, 12.78446592408340, 13.18532536097134, 13.85534190091318