| L(s) = 1 | + 2-s − 1.02·3-s + 4-s + 1.43·5-s − 1.02·6-s + 5.08·7-s + 8-s − 1.95·9-s + 1.43·10-s + 4.14·11-s − 1.02·12-s + 5.08·14-s − 1.46·15-s + 16-s + 5.61·17-s − 1.95·18-s − 19-s + 1.43·20-s − 5.18·21-s + 4.14·22-s + 4.40·23-s − 1.02·24-s − 2.93·25-s + 5.06·27-s + 5.08·28-s − 4.53·29-s − 1.46·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.589·3-s + 0.5·4-s + 0.643·5-s − 0.416·6-s + 1.92·7-s + 0.353·8-s − 0.652·9-s + 0.454·10-s + 1.25·11-s − 0.294·12-s + 1.35·14-s − 0.379·15-s + 0.250·16-s + 1.36·17-s − 0.461·18-s − 0.229·19-s + 0.321·20-s − 1.13·21-s + 0.884·22-s + 0.919·23-s − 0.208·24-s − 0.586·25-s + 0.974·27-s + 0.960·28-s − 0.842·29-s − 0.268·30-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
| \(L(1)\) |
\(\approx\) |
\(4.261989396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.261989396\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 4.53T + 29T^{2} \) |
| 31 | \( 1 + 6.77T + 31T^{2} \) |
| 37 | \( 1 - 7.83T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 7.47T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 3.27T + 83T^{2} \) |
| 89 | \( 1 - 3.71T + 89T^{2} \) |
| 97 | \( 1 - 0.438T + 97T^{2} \) |
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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
−7.71007318640055607875681227953, −7.41995163149906498258560869679, −6.29802438536881353740225396923, −5.65446544304847212912093953161, −5.38043798456055596596671911824, −4.50755168094159054028093958596, −3.84941128664574634385955234586, −2.73088027680372069563556020817, −1.72544632810094487622656635190, −1.12478457368260616816943438819,
1.12478457368260616816943438819, 1.72544632810094487622656635190, 2.73088027680372069563556020817, 3.84941128664574634385955234586, 4.50755168094159054028093958596, 5.38043798456055596596671911824, 5.65446544304847212912093953161, 6.29802438536881353740225396923, 7.41995163149906498258560869679, 7.71007318640055607875681227953
