| L(s) = 1 | − 2·2-s + 24.9·3-s − 28·4-s + 74.9·5-s − 49.9·6-s + 120·8-s + 381·9-s − 149.·10-s − 284·11-s − 699.·12-s + 524.·13-s + 1.87e3·15-s + 656·16-s − 149.·17-s − 762·18-s + 2.17e3·19-s − 2.09e3·20-s + 568·22-s + 1.49e3·23-s + 2.99e3·24-s + 2.49e3·25-s − 1.04e3·26-s + 3.44e3·27-s − 4.36e3·29-s − 3.74e3·30-s − 6.44e3·31-s − 5.15e3·32-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.60·3-s − 0.875·4-s + 1.34·5-s − 0.566·6-s + 0.662·8-s + 1.56·9-s − 0.473·10-s − 0.707·11-s − 1.40·12-s + 0.860·13-s + 2.14·15-s + 0.640·16-s − 0.125·17-s − 0.554·18-s + 1.38·19-s − 1.17·20-s + 0.250·22-s + 0.589·23-s + 1.06·24-s + 0.797·25-s − 0.304·26-s + 0.910·27-s − 0.964·29-s − 0.759·30-s − 1.20·31-s − 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.346613588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.346613588\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 2T + 32T^{2} \) |
| 3 | \( 1 - 24.9T + 243T^{2} \) |
| 5 | \( 1 - 74.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 284T + 1.61e5T^{2} \) |
| 13 | \( 1 - 524.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 149.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.36e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.64e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.07e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 524.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.83e5T + 8.58e9T^{2} \) |
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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
−14.19011838591773733281930931658, −13.64143055369740617567370190390, −12.99491928948682677511876178260, −10.45225962195148426676160884157, −9.412060224854830657200647054635, −8.817329721601413741160473146994, −7.53713322833979450129283956013, −5.36834838610195200544384892644, −3.39528302569368729617318245128, −1.69132044578871428722533862179,
1.69132044578871428722533862179, 3.39528302569368729617318245128, 5.36834838610195200544384892644, 7.53713322833979450129283956013, 8.817329721601413741160473146994, 9.412060224854830657200647054635, 10.45225962195148426676160884157, 12.99491928948682677511876178260, 13.64143055369740617567370190390, 14.19011838591773733281930931658
