| L(s) = 1 | − 2-s + 1.67·3-s + 4-s − 2.31·5-s − 1.67·6-s + 0.643·7-s − 8-s − 0.181·9-s + 2.31·10-s − 11-s + 1.67·12-s − 1.92·13-s − 0.643·14-s − 3.88·15-s + 16-s + 4.51·17-s + 0.181·18-s + 2.52·19-s − 2.31·20-s + 1.08·21-s + 22-s + 5.22·23-s − 1.67·24-s + 0.357·25-s + 1.92·26-s − 5.34·27-s + 0.643·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.969·3-s + 0.5·4-s − 1.03·5-s − 0.685·6-s + 0.243·7-s − 0.353·8-s − 0.0606·9-s + 0.731·10-s − 0.301·11-s + 0.484·12-s − 0.533·13-s − 0.171·14-s − 1.00·15-s + 0.250·16-s + 1.09·17-s + 0.0428·18-s + 0.579·19-s − 0.517·20-s + 0.235·21-s + 0.213·22-s + 1.08·23-s − 0.342·24-s + 0.0715·25-s + 0.377·26-s − 1.02·27-s + 0.121·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 - 0.643T + 7T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 2.52T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 0.0236T + 31T^{2} \) |
| 37 | \( 1 + 7.10T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 - 0.191T + 43T^{2} \) |
| 47 | \( 1 - 1.10T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 59 | \( 1 + 3.43T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 4.52T + 97T^{2} \) |
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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
−7.978154922702799681970934801919, −7.53404341762167652861496859376, −7.08115977869400246880650447778, −5.78201983448661713473850200891, −5.05534460163024645873588983474, −3.90705145952495292462714013580, −3.23622636196062907951055176408, −2.54452408151974771558191300301, −1.36072762991644874515097963593, 0,
1.36072762991644874515097963593, 2.54452408151974771558191300301, 3.23622636196062907951055176408, 3.90705145952495292462714013580, 5.05534460163024645873588983474, 5.78201983448661713473850200891, 7.08115977869400246880650447778, 7.53404341762167652861496859376, 7.978154922702799681970934801919
