| L(s) = 1 | − 3.71·2-s − 24.7·3-s − 114.·4-s + 385.·5-s + 91.9·6-s − 1.54e3·7-s + 900.·8-s − 1.57e3·9-s − 1.43e3·10-s − 1.66e3·11-s + 2.82e3·12-s + 1.27e4·13-s + 5.74e3·14-s − 9.53e3·15-s + 1.12e4·16-s + 2.38e4·17-s + 5.85e3·18-s + 3.05e4·19-s − 4.40e4·20-s + 3.82e4·21-s + 6.17e3·22-s − 6.37e4·23-s − 2.22e4·24-s + 7.03e4·25-s − 4.72e4·26-s + 9.30e4·27-s + 1.76e5·28-s + ⋯ |
| L(s) = 1 | − 0.328·2-s − 0.529·3-s − 0.892·4-s + 1.37·5-s + 0.173·6-s − 1.70·7-s + 0.621·8-s − 0.719·9-s − 0.452·10-s − 0.376·11-s + 0.472·12-s + 1.60·13-s + 0.559·14-s − 0.729·15-s + 0.687·16-s + 1.17·17-s + 0.236·18-s + 1.02·19-s − 1.22·20-s + 0.901·21-s + 0.123·22-s − 1.09·23-s − 0.328·24-s + 0.900·25-s − 0.527·26-s + 0.910·27-s + 1.52·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.002401660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002401660\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 + 3.71T + 128T^{2} \) |
| 3 | \( 1 + 24.7T + 2.18e3T^{2} \) |
| 5 | \( 1 - 385.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.54e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.27e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.38e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.05e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.90e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.43e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.38e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.55e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 3.94e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.24e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 7.51e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.22e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.63e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.49e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 8.46e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.99e6T + 8.07e13T^{2} \) |
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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
−13.84632877057426408243208177945, −13.57820776238712133350412394739, −12.25883083815280489930045941790, −10.30615329754780594206236211774, −9.744377437670068081877593558939, −8.515800199076012361123677721784, −6.27581306900272409262278027635, −5.54155410768436745706398461625, −3.25902929607434875731701060843, −0.831624383573782639258420854093,
0.831624383573782639258420854093, 3.25902929607434875731701060843, 5.54155410768436745706398461625, 6.27581306900272409262278027635, 8.515800199076012361123677721784, 9.744377437670068081877593558939, 10.30615329754780594206236211774, 12.25883083815280489930045941790, 13.57820776238712133350412394739, 13.84632877057426408243208177945
