L(s) = 1 | − 2-s + 2.71·3-s + 4-s + 0.231·5-s − 2.71·6-s + 3.53·7-s − 8-s + 4.36·9-s − 0.231·10-s + 4.96·11-s + 2.71·12-s − 3.53·14-s + 0.628·15-s + 16-s + 0.380·17-s − 4.36·18-s − 19-s + 0.231·20-s + 9.59·21-s − 4.96·22-s − 8.03·23-s − 2.71·24-s − 4.94·25-s + 3.71·27-s + 3.53·28-s + 3.83·29-s − 0.628·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.56·3-s + 0.5·4-s + 0.103·5-s − 1.10·6-s + 1.33·7-s − 0.353·8-s + 1.45·9-s − 0.0732·10-s + 1.49·11-s + 0.783·12-s − 0.944·14-s + 0.162·15-s + 0.250·16-s + 0.0923·17-s − 1.02·18-s − 0.229·19-s + 0.0517·20-s + 2.09·21-s − 1.05·22-s − 1.67·23-s − 0.554·24-s − 0.989·25-s + 0.714·27-s + 0.668·28-s + 0.712·29-s − 0.114·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.620969109\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.620969109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 - 0.231T + 5T^{2} \) |
| 7 | \( 1 - 3.53T + 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 17 | \( 1 - 0.380T + 17T^{2} \) |
| 23 | \( 1 + 8.03T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 37 | \( 1 + 0.944T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 - 0.0285T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6.70T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 0.0332T + 79T^{2} \) |
| 83 | \( 1 - 0.0511T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−8.111127239158722505562887405806, −7.79162860564644013156401347466, −6.78575516603276885803460612595, −6.21382850041990856541684094144, −5.02724551908243184111742427499, −4.07690113982428945569596946547, −3.64737769042959566232799051959, −2.41210968793806971633398338458, −1.91303544280726526224181415961, −1.11226274979288374967981199420,
1.11226274979288374967981199420, 1.91303544280726526224181415961, 2.41210968793806971633398338458, 3.64737769042959566232799051959, 4.07690113982428945569596946547, 5.02724551908243184111742427499, 6.21382850041990856541684094144, 6.78575516603276885803460612595, 7.79162860564644013156401347466, 8.111127239158722505562887405806