| L(s) = 1 | + 2.53·3-s + 0.138·5-s + 7-s + 3.44·9-s − 0.938·11-s + 3.02·13-s + 0.352·15-s + 17-s − 2.38·19-s + 2.53·21-s + 5.51·23-s − 4.98·25-s + 1.11·27-s + 4.99·29-s + 9.85·31-s − 2.38·33-s + 0.138·35-s + 6.85·37-s + 7.68·39-s − 4.61·41-s + 1.86·43-s + 0.477·45-s − 0.273·47-s + 49-s + 2.53·51-s − 6.18·53-s − 0.130·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 0.0621·5-s + 0.377·7-s + 1.14·9-s − 0.282·11-s + 0.840·13-s + 0.0910·15-s + 0.242·17-s − 0.546·19-s + 0.553·21-s + 1.15·23-s − 0.996·25-s + 0.215·27-s + 0.927·29-s + 1.76·31-s − 0.414·33-s + 0.0234·35-s + 1.12·37-s + 1.23·39-s − 0.720·41-s + 0.283·43-s + 0.0712·45-s − 0.0398·47-s + 0.142·49-s + 0.355·51-s − 0.849·53-s − 0.0175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.157247499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.157247499\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 0.138T + 5T^{2} \) |
| 11 | \( 1 + 0.938T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 - 5.51T + 23T^{2} \) |
| 29 | \( 1 - 4.99T + 29T^{2} \) |
| 31 | \( 1 - 9.85T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + 0.273T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 - 2.88T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 - 5.80T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.96T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.011949463746335067777884286521, −7.46175913510353003737218760732, −6.54132148544838753143293968432, −5.90483412958623464510027315952, −4.82136475998266345952078748498, −4.23738219663799888934036035791, −3.33832610050524610853350535886, −2.77580241130337242987031403885, −1.95848291349599254021187107662, −0.994927287576462105406171052714,
0.994927287576462105406171052714, 1.95848291349599254021187107662, 2.77580241130337242987031403885, 3.33832610050524610853350535886, 4.23738219663799888934036035791, 4.82136475998266345952078748498, 5.90483412958623464510027315952, 6.54132148544838753143293968432, 7.46175913510353003737218760732, 8.011949463746335067777884286521
