| L(s) = 1 | − 3-s − 2·4-s − 5-s − 2·7-s + 9-s + 11-s + 2·12-s + 6·13-s + 15-s + 4·16-s + 4·17-s − 2·19-s + 2·20-s + 2·21-s + 3·23-s + 25-s − 27-s + 4·28-s + 29-s − 4·31-s − 33-s + 2·35-s − 2·36-s − 3·37-s − 6·39-s + 7·41-s + 5·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 1.66·13-s + 0.258·15-s + 16-s + 0.970·17-s − 0.458·19-s + 0.447·20-s + 0.436·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.755·28-s + 0.185·29-s − 0.718·31-s − 0.174·33-s + 0.338·35-s − 1/3·36-s − 0.493·37-s − 0.960·39-s + 1.09·41-s + 0.762·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8094965276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8094965276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−11.04916167654196644301100657525, −10.31332587086161814362537835961, −9.256030022852843175336459161061, −8.593420838968236541812101317012, −7.46677992367070753727549366405, −6.26802946364034461238820028257, −5.47980721520530440715156681592, −4.16721175306965324518809801959, −3.43182364531126383498212067183, −0.911659172267844473526681142024,
0.911659172267844473526681142024, 3.43182364531126383498212067183, 4.16721175306965324518809801959, 5.47980721520530440715156681592, 6.26802946364034461238820028257, 7.46677992367070753727549366405, 8.593420838968236541812101317012, 9.256030022852843175336459161061, 10.31332587086161814362537835961, 11.04916167654196644301100657525
