| L(s) = 1 | − 2-s − 3-s + 4-s + 0.780·5-s + 6-s + 1.13·7-s − 8-s + 9-s − 0.780·10-s − 3.87·11-s − 12-s + 0.838·13-s − 1.13·14-s − 0.780·15-s + 16-s + 17-s − 18-s − 5.40·19-s + 0.780·20-s − 1.13·21-s + 3.87·22-s + 0.359·23-s + 24-s − 4.39·25-s − 0.838·26-s − 27-s + 1.13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.349·5-s + 0.408·6-s + 0.429·7-s − 0.353·8-s + 0.333·9-s − 0.246·10-s − 1.16·11-s − 0.288·12-s + 0.232·13-s − 0.303·14-s − 0.201·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.24·19-s + 0.174·20-s − 0.248·21-s + 0.826·22-s + 0.0749·23-s + 0.204·24-s − 0.878·25-s − 0.164·26-s − 0.192·27-s + 0.214·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9399027836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9399027836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 0.780T + 5T^{2} \) |
| 7 | \( 1 - 1.13T + 7T^{2} \) |
| 11 | \( 1 + 3.87T + 11T^{2} \) |
| 13 | \( 1 - 0.838T + 13T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 0.359T + 23T^{2} \) |
| 29 | \( 1 + 1.02T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 0.228T + 37T^{2} \) |
| 41 | \( 1 + 2.52T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 7.64T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 - 5.59T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 6.73T + 89T^{2} \) |
| 97 | \( 1 - 6.10T + 97T^{2} \) |
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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
−8.257830382935453542871483243280, −7.49237647524142784894971735146, −6.56758234435104615104051125808, −6.13366523827792027409294672368, −5.23524012298817914284838132595, −4.68003362509704731223788852135, −3.57516229014449973682854396047, −2.47531346497892199187740782135, −1.77643119847789249150777787596, −0.57538661755357684507457007619,
0.57538661755357684507457007619, 1.77643119847789249150777787596, 2.47531346497892199187740782135, 3.57516229014449973682854396047, 4.68003362509704731223788852135, 5.23524012298817914284838132595, 6.13366523827792027409294672368, 6.56758234435104615104051125808, 7.49237647524142784894971735146, 8.257830382935453542871483243280
