| L(s) = 1 | + 64·2-s + 1.95e3·3-s + 4.09e3·4-s + 4.87e4·5-s + 1.25e5·6-s − 1.17e5·7-s + 2.62e5·8-s + 2.24e6·9-s + 3.11e6·10-s − 8.07e6·11-s + 8.02e6·12-s − 1.35e7·13-s − 7.52e6·14-s + 9.54e7·15-s + 1.67e7·16-s − 8.11e7·17-s + 1.43e8·18-s + 2.14e8·19-s + 1.99e8·20-s − 2.30e8·21-s − 5.16e8·22-s + 8.24e8·23-s + 5.13e8·24-s + 1.15e9·25-s − 8.67e8·26-s + 1.26e9·27-s − 4.81e8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.55·3-s + 0.5·4-s + 1.39·5-s + 1.09·6-s − 0.377·7-s + 0.353·8-s + 1.40·9-s + 0.986·10-s − 1.37·11-s + 0.775·12-s − 0.779·13-s − 0.267·14-s + 2.16·15-s + 0.250·16-s − 0.815·17-s + 0.993·18-s + 1.04·19-s + 0.697·20-s − 0.586·21-s − 0.971·22-s + 1.16·23-s + 0.548·24-s + 0.944·25-s − 0.550·26-s + 0.628·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(4.986361304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.986361304\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 64T \) |
| 7 | \( 1 + 1.17e5T \) |
| good | 3 | \( 1 - 1.95e3T + 1.59e6T^{2} \) |
| 5 | \( 1 - 4.87e4T + 1.22e9T^{2} \) |
| 11 | \( 1 + 8.07e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.35e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 8.11e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.14e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 8.24e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.87e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 8.11e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 3.00e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 1.26e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.88e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 6.60e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.60e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 4.22e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 3.96e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 1.27e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.11e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.52e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.21e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 7.62e11T + 8.87e24T^{2} \) |
| 89 | \( 1 - 6.86e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.92e12T + 6.73e25T^{2} \) |
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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
−15.89462744844159834780368772918, −14.62211936899021088037447848010, −13.60729630966934360986294781683, −12.92535370025895523305839901885, −10.28638481061411245564331119458, −9.068068546088278185318342910768, −7.24720152132229635850519590283, −5.25084144316708502958802193198, −3.02330153581169147138969789792, −2.09272711639960391715373432473,
2.09272711639960391715373432473, 3.02330153581169147138969789792, 5.25084144316708502958802193198, 7.24720152132229635850519590283, 9.068068546088278185318342910768, 10.28638481061411245564331119458, 12.92535370025895523305839901885, 13.60729630966934360986294781683, 14.62211936899021088037447848010, 15.89462744844159834780368772918
