| L(s) = 1 | − 6.09·2-s + 250.·3-s − 474.·4-s − 546.·5-s − 1.52e3·6-s + 6.56e3·7-s + 6.01e3·8-s + 4.32e4·9-s + 3.32e3·10-s − 3.98e4·11-s − 1.19e5·12-s + 1.76e5·13-s − 4.00e4·14-s − 1.36e5·15-s + 2.06e5·16-s − 3.95e5·17-s − 2.63e5·18-s + 7.99e5·19-s + 2.59e5·20-s + 1.64e6·21-s + 2.42e5·22-s + 2.46e6·23-s + 1.50e6·24-s − 1.65e6·25-s − 1.07e6·26-s + 5.91e6·27-s − 3.11e6·28-s + ⋯ |
| L(s) = 1 | − 0.269·2-s + 1.78·3-s − 0.927·4-s − 0.390·5-s − 0.481·6-s + 1.03·7-s + 0.519·8-s + 2.19·9-s + 0.105·10-s − 0.819·11-s − 1.65·12-s + 1.71·13-s − 0.278·14-s − 0.698·15-s + 0.787·16-s − 1.14·17-s − 0.591·18-s + 1.40·19-s + 0.362·20-s + 1.84·21-s + 0.220·22-s + 1.83·23-s + 0.928·24-s − 0.847·25-s − 0.461·26-s + 2.14·27-s − 0.958·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.611828131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.611828131\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 7.07e5T \) |
| good | 2 | \( 1 + 6.09T + 512T^{2} \) |
| 3 | \( 1 - 250.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 546.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.98e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.76e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.95e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.99e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.46e6T + 1.80e12T^{2} \) |
| 31 | \( 1 - 1.40e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.13e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.84e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 7.67e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.72e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.10e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.31e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.46e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.84e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.84e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.09e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.14e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.99e8T + 7.60e17T^{2} \) |
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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
−14.91283356910497129383575235401, −13.70129753617459577385990770055, −13.24009820997923672540156355178, −10.97355362647358996021706226365, −9.323883838931871465871179093328, −8.437271833885169215264528336763, −7.68613985437475188419823746407, −4.66904485594666276116701117048, −3.30200023961995763625908690873, −1.36882052285547329693874202998,
1.36882052285547329693874202998, 3.30200023961995763625908690873, 4.66904485594666276116701117048, 7.68613985437475188419823746407, 8.437271833885169215264528336763, 9.323883838931871465871179093328, 10.97355362647358996021706226365, 13.24009820997923672540156355178, 13.70129753617459577385990770055, 14.91283356910497129383575235401
