| L(s) = 1 | − 8.56·2-s + 18.3·3-s + 41.4·4-s − 84.3·5-s − 157.·6-s + 216.·7-s − 80.7·8-s + 92.8·9-s + 722.·10-s + 380.·11-s + 759.·12-s + 1.05e3·13-s − 1.85e3·14-s − 1.54e3·15-s − 633.·16-s + 801.·17-s − 796.·18-s − 288.·19-s − 3.49e3·20-s + 3.97e3·21-s − 3.26e3·22-s + 334.·23-s − 1.48e3·24-s + 3.98e3·25-s − 9.06e3·26-s − 2.75e3·27-s + 8.98e3·28-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.17·3-s + 1.29·4-s − 1.50·5-s − 1.78·6-s + 1.67·7-s − 0.446·8-s + 0.382·9-s + 2.28·10-s + 0.948·11-s + 1.52·12-s + 1.73·13-s − 2.53·14-s − 1.77·15-s − 0.618·16-s + 0.672·17-s − 0.579·18-s − 0.183·19-s − 1.95·20-s + 1.96·21-s − 1.43·22-s + 0.131·23-s − 0.524·24-s + 1.27·25-s − 2.63·26-s − 0.726·27-s + 2.16·28-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
| \(L(3)\) |
\(\approx\) |
\(1.030579302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030579302\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 8.56T + 32T^{2} \) |
| 3 | \( 1 - 18.3T + 243T^{2} \) |
| 5 | \( 1 + 84.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 216.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 380.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 801.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 288.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 334.T + 6.43e6T^{2} \) |
| 31 | \( 1 + 3.28e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 887.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.32e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.33e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.10e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.62e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.70e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.33e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.46e4T + 8.58e9T^{2} \) |
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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
−16.13526098147572520445268684889, −15.05023524382385714679617882096, −14.03535332444360206452457610918, −11.68413779025094828304757179304, −10.90439766776959295497308586327, −8.826833099525399741526973092963, −8.353533238697921362451917827713, −7.47182427436995463457838168086, −3.89385227202194829994886489466, −1.36422687186964175748597732311,
1.36422687186964175748597732311, 3.89385227202194829994886489466, 7.47182427436995463457838168086, 8.353533238697921362451917827713, 8.826833099525399741526973092963, 10.90439766776959295497308586327, 11.68413779025094828304757179304, 14.03535332444360206452457610918, 15.05023524382385714679617882096, 16.13526098147572520445268684889
