L(s) = 1 | − 0.925·2-s − 3-s − 1.14·4-s + 3.08·5-s + 0.925·6-s − 0.0396·7-s + 2.90·8-s + 9-s − 2.85·10-s + 3.49·11-s + 1.14·12-s + 13-s + 0.0366·14-s − 3.08·15-s − 0.403·16-s + 4.97·17-s − 0.925·18-s + 5.15·19-s − 3.53·20-s + 0.0396·21-s − 3.23·22-s + 5.51·23-s − 2.90·24-s + 4.54·25-s − 0.925·26-s − 27-s + 0.0453·28-s + ⋯ |
L(s) = 1 | − 0.654·2-s − 0.577·3-s − 0.571·4-s + 1.38·5-s + 0.377·6-s − 0.0149·7-s + 1.02·8-s + 0.333·9-s − 0.904·10-s + 1.05·11-s + 0.330·12-s + 0.277·13-s + 0.00979·14-s − 0.797·15-s − 0.100·16-s + 1.20·17-s − 0.218·18-s + 1.18·19-s − 0.790·20-s + 0.00864·21-s − 0.689·22-s + 1.15·23-s − 0.593·24-s + 0.909·25-s − 0.181·26-s − 0.192·27-s + 0.00856·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.637447302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637447302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.925T + 2T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 + 0.0396T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 8.28T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.138T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 1.88T + 61T^{2} \) |
| 67 | \( 1 - 5.01T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 7.21T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 2.35T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.663500803321032443179413064934, −7.69944218597294205043499632345, −7.06139294438915319291800800734, −6.04066829803343758531104720296, −5.65219496524195166723949261290, −4.83468986208413597638411446500, −3.95857036873400022508962609929, −2.83307088916493468882975203937, −1.40210670130232106700090785052, −1.02267452900582628883960278985,
1.02267452900582628883960278985, 1.40210670130232106700090785052, 2.83307088916493468882975203937, 3.95857036873400022508962609929, 4.83468986208413597638411446500, 5.65219496524195166723949261290, 6.04066829803343758531104720296, 7.06139294438915319291800800734, 7.69944218597294205043499632345, 8.663500803321032443179413064934