| L(s) = 1 | + 1.69·2-s + 56.8·3-s − 125.·4-s + 194.·5-s + 96.3·6-s + 1.22e3·7-s − 429.·8-s + 1.04e3·9-s + 330.·10-s + 3.77e3·11-s − 7.11e3·12-s − 481.·13-s + 2.07e3·14-s + 1.10e4·15-s + 1.52e4·16-s + 2.47e4·17-s + 1.77e3·18-s + 6.94e3·19-s − 2.43e4·20-s + 6.94e4·21-s + 6.40e3·22-s − 5.08e4·23-s − 2.43e4·24-s − 4.01e4·25-s − 816.·26-s − 6.48e4·27-s − 1.52e5·28-s + ⋯ |
| L(s) = 1 | + 0.149·2-s + 1.21·3-s − 0.977·4-s + 0.697·5-s + 0.182·6-s + 1.34·7-s − 0.296·8-s + 0.478·9-s + 0.104·10-s + 0.855·11-s − 1.18·12-s − 0.0608·13-s + 0.201·14-s + 0.848·15-s + 0.933·16-s + 1.21·17-s + 0.0717·18-s + 0.232·19-s − 0.681·20-s + 1.63·21-s + 0.128·22-s − 0.871·23-s − 0.360·24-s − 0.513·25-s − 0.00911·26-s − 0.633·27-s − 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.705482779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.705482779\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.43e4T \) |
| good | 2 | \( 1 - 1.69T + 128T^{2} \) |
| 3 | \( 1 - 56.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 194.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.22e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.77e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 481.T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.47e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.94e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.08e4T + 3.40e9T^{2} \) |
| 31 | \( 1 + 1.01e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.96e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.64e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.55e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.99e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.61e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.06e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.83e6T + 8.07e13T^{2} \) |
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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
−14.88238241346939002591240516764, −14.21954606313420539490403915989, −13.63417194691792519835562014276, −11.96775554595301485120973221965, −9.955566660937079347238320414982, −8.867733755674445405364561080699, −7.88833074360023829529887236059, −5.40302878025778307221133317908, −3.73472054143053654216144009592, −1.67602620536339163202779569201,
1.67602620536339163202779569201, 3.73472054143053654216144009592, 5.40302878025778307221133317908, 7.88833074360023829529887236059, 8.867733755674445405364561080699, 9.955566660937079347238320414982, 11.96775554595301485120973221965, 13.63417194691792519835562014276, 14.21954606313420539490403915989, 14.88238241346939002591240516764
