| L(s) = 1 | − 6.83·2-s − 24.5·3-s + 14.7·4-s − 54.3·5-s + 167.·6-s − 37.6·7-s + 117.·8-s + 360.·9-s + 371.·10-s − 146.·11-s − 362.·12-s + 162.·13-s + 257.·14-s + 1.33e3·15-s − 1.27e3·16-s − 2.16e3·17-s − 2.46e3·18-s + 2.49e3·19-s − 801.·20-s + 925.·21-s + 1.00e3·22-s − 122.·23-s − 2.89e3·24-s − 175.·25-s − 1.11e3·26-s − 2.88e3·27-s − 556.·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 1.57·3-s + 0.461·4-s − 0.971·5-s + 1.90·6-s − 0.290·7-s + 0.651·8-s + 1.48·9-s + 1.17·10-s − 0.366·11-s − 0.727·12-s + 0.266·13-s + 0.351·14-s + 1.53·15-s − 1.24·16-s − 1.81·17-s − 1.79·18-s + 1.58·19-s − 0.448·20-s + 0.457·21-s + 0.442·22-s − 0.0483·23-s − 1.02·24-s − 0.0562·25-s − 0.322·26-s − 0.761·27-s − 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2446683834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2446683834\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 841T \) |
| good | 2 | \( 1 + 6.83T + 32T^{2} \) |
| 3 | \( 1 + 24.5T + 243T^{2} \) |
| 5 | \( 1 + 54.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 37.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 146.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 162.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 122.T + 6.43e6T^{2} \) |
| 31 | \( 1 - 8.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.17e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.51e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.82e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.39e4T + 8.58e9T^{2} \) |
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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
−16.18753829456703183570821871626, −15.77934671429104660339014950027, −13.27017360244648197793714844445, −11.69122026557819540638580946656, −11.02245534148911932220576541323, −9.708132633585089077242493818204, −8.006758632975077432369791712368, −6.62522364837222268506810619597, −4.66242522013045526807422299379, −0.60058886752604815275427474811,
0.60058886752604815275427474811, 4.66242522013045526807422299379, 6.62522364837222268506810619597, 8.006758632975077432369791712368, 9.708132633585089077242493818204, 11.02245534148911932220576541323, 11.69122026557819540638580946656, 13.27017360244648197793714844445, 15.77934671429104660339014950027, 16.18753829456703183570821871626
