L(s) = 1 | + 3-s + 5-s + 9-s + 6·11-s + 2·13-s + 15-s + 4·17-s − 2·19-s + 23-s + 25-s + 27-s − 8·29-s − 6·31-s + 6·33-s + 2·37-s + 2·39-s + 10·41-s − 2·43-s + 45-s + 2·47-s + 4·51-s − 6·53-s + 6·55-s − 2·57-s − 4·59-s − 6·61-s + 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.258·15-s + 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.07·31-s + 1.04·33-s + 0.328·37-s + 0.320·39-s + 1.56·41-s − 0.304·43-s + 0.149·45-s + 0.291·47-s + 0.560·51-s − 0.824·53-s + 0.809·55-s − 0.264·57-s − 0.520·59-s − 0.768·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.943532630\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.943532630\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.56604831032029, −12.88459530826371, −12.56574833929171, −12.16942949244580, −11.36512892709943, −11.11421205995024, −10.63805769552789, −9.810300907010404, −9.492759321955541, −9.147228040139341, −8.737558667969081, −8.063637910959680, −7.568240983677638, −7.060478225665599, −6.487413516411219, −5.984600575437028, −5.609156592545998, −4.810022958088142, −4.157419165444472, −3.691644474364652, −3.337151927439430, −2.507774065553963, −1.763984651654107, −1.418523382400056, −0.6565202749996864,
0.6565202749996864, 1.418523382400056, 1.763984651654107, 2.507774065553963, 3.337151927439430, 3.691644474364652, 4.157419165444472, 4.810022958088142, 5.609156592545998, 5.984600575437028, 6.487413516411219, 7.060478225665599, 7.568240983677638, 8.063637910959680, 8.737558667969081, 9.147228040139341, 9.492759321955541, 9.810300907010404, 10.63805769552789, 11.11421205995024, 11.36512892709943, 12.16942949244580, 12.56574833929171, 12.88459530826371, 13.56604831032029