L(s) = 1 | + 62.2·2-s − 384.·3-s + 1.83e3·4-s + 1.37e3·5-s − 2.39e4·6-s + 3.30e4·7-s − 1.35e4·8-s − 2.91e4·9-s + 8.57e4·10-s − 8.00e5·11-s − 7.03e5·12-s + 1.48e6·13-s + 2.05e6·14-s − 5.29e5·15-s − 4.59e6·16-s − 1.13e7·17-s − 1.81e6·18-s − 9.78e6·19-s + 2.52e6·20-s − 1.27e7·21-s − 4.98e7·22-s + 5.46e5·23-s + 5.22e6·24-s − 4.69e7·25-s + 9.25e7·26-s + 7.93e7·27-s + 6.04e7·28-s + ⋯ |
L(s) = 1 | + 1.37·2-s − 0.913·3-s + 0.893·4-s + 0.197·5-s − 1.25·6-s + 0.743·7-s − 0.146·8-s − 0.164·9-s + 0.271·10-s − 1.49·11-s − 0.816·12-s + 1.10·13-s + 1.02·14-s − 0.180·15-s − 1.09·16-s − 1.93·17-s − 0.226·18-s − 0.906·19-s + 0.176·20-s − 0.679·21-s − 2.06·22-s + 0.0176·23-s + 0.133·24-s − 0.961·25-s + 1.52·26-s + 1.06·27-s + 0.664·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 - 2.05e7T \) |
good | 2 | \( 1 - 62.2T + 2.04e3T^{2} \) |
| 3 | \( 1 + 384.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.37e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.30e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.00e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.48e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.13e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.78e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.46e5T + 9.52e14T^{2} \) |
| 31 | \( 1 - 4.17e6T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.50e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.34e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.74e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.70e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.30e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.84e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.13e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.85e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.82e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.81e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.94e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.27e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.46e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.40e10T + 7.15e21T^{2} \) |
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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
−13.66503578164396657718120411603, −12.94007247475097494298616766339, −11.48245844543128752891157656516, −10.80941303176049225843541504190, −8.486070669408914160620511493035, −6.40530109529597682147850237669, −5.41267843264848241503571637047, −4.33908871653396548875867134884, −2.38301702766475081334257295754, 0,
2.38301702766475081334257295754, 4.33908871653396548875867134884, 5.41267843264848241503571637047, 6.40530109529597682147850237669, 8.486070669408914160620511493035, 10.80941303176049225843541504190, 11.48245844543128752891157656516, 12.94007247475097494298616766339, 13.66503578164396657718120411603