| L(s) = 1 | − 15.5·2-s − 64.5·3-s + 115.·4-s − 389.·5-s + 1.00e3·6-s + 1.43e3·7-s + 200.·8-s + 1.98e3·9-s + 6.06e3·10-s − 7.43e3·12-s − 9.86e3·13-s − 2.24e4·14-s + 2.51e4·15-s − 1.78e4·16-s + 1.28e4·17-s − 3.09e4·18-s − 2.94e4·19-s − 4.48e4·20-s − 9.29e4·21-s − 1.31e4·23-s − 1.29e4·24-s + 7.33e4·25-s + 1.53e5·26-s + 1.30e4·27-s + 1.65e5·28-s − 8.06e4·29-s − 3.92e5·30-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 1.38·3-s + 0.899·4-s − 1.39·5-s + 1.90·6-s + 1.58·7-s + 0.138·8-s + 0.907·9-s + 1.91·10-s − 1.24·12-s − 1.24·13-s − 2.18·14-s + 1.92·15-s − 1.09·16-s + 0.633·17-s − 1.25·18-s − 0.985·19-s − 1.25·20-s − 2.19·21-s − 0.226·23-s − 0.191·24-s + 0.939·25-s + 1.71·26-s + 0.127·27-s + 1.42·28-s − 0.614·29-s − 2.65·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1646536987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1646536987\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 15.5T + 128T^{2} \) |
| 3 | \( 1 + 64.5T + 2.18e3T^{2} \) |
| 5 | \( 1 + 389.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.43e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 9.86e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.28e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.94e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.31e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.06e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.23e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.06e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.07e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.50e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.01e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.19e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 7.42e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.64e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.13e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.02e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.29e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.53e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.81e5T + 8.07e13T^{2} \) |
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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
−11.65679243397772052618537740386, −11.05692784602064450193334662499, −10.28969531394535755549958191146, −8.716518216228412184831857209497, −7.77385993220177147356270538159, −7.13205300392413729912129106712, −5.25846035797423090658052567970, −4.33131698808915869606595720049, −1.68856544906191744912013651679, −0.32428342762098731747886361335,
0.32428342762098731747886361335, 1.68856544906191744912013651679, 4.33131698808915869606595720049, 5.25846035797423090658052567970, 7.13205300392413729912129106712, 7.77385993220177147356270538159, 8.716518216228412184831857209497, 10.28969531394535755549958191146, 11.05692784602064450193334662499, 11.65679243397772052618537740386
