| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2·7-s − 3·8-s + 9-s − 6·11-s + 12-s + 2·14-s − 16-s + 6·17-s + 18-s + 19-s − 2·21-s − 6·22-s + 8·23-s + 3·24-s − 27-s − 2·28-s + 4·29-s + 5·32-s + 6·33-s + 6·34-s − 36-s − 4·37-s + 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.534·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.436·21-s − 1.27·22-s + 1.66·23-s + 0.612·24-s − 0.192·27-s − 0.377·28-s + 0.742·29-s + 0.883·32-s + 1.04·33-s + 1.02·34-s − 1/6·36-s − 0.657·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.578349064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578349064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.698951273659044584177084077294, −8.612427267322374531961751053288, −7.924047454743910880914084945863, −7.09516569151059227985278865024, −5.84081207636873725415960197609, −5.12941821039507935665099026792, −4.90521166564327801764556289537, −3.58795054204413648892600902189, −2.62332567549261815611306089460, −0.851831854705753066065028722173,
0.851831854705753066065028722173, 2.62332567549261815611306089460, 3.58795054204413648892600902189, 4.90521166564327801764556289537, 5.12941821039507935665099026792, 5.84081207636873725415960197609, 7.09516569151059227985278865024, 7.924047454743910880914084945863, 8.612427267322374531961751053288, 9.698951273659044584177084077294
