| L(s) = 1 | − 5.05·2-s − 9.64·3-s + 17.5·4-s + 12.9·5-s + 48.7·6-s − 7·7-s − 48.4·8-s + 66.0·9-s − 65.4·10-s + 11·11-s − 169.·12-s − 55.5·13-s + 35.3·14-s − 124.·15-s + 104.·16-s + 59.2·17-s − 333.·18-s − 33.1·19-s + 227.·20-s + 67.5·21-s − 55.6·22-s + 26.0·23-s + 466.·24-s + 42.5·25-s + 280.·26-s − 376.·27-s − 123.·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 1.85·3-s + 2.19·4-s + 1.15·5-s + 3.31·6-s − 0.377·7-s − 2.13·8-s + 2.44·9-s − 2.06·10-s + 0.301·11-s − 4.07·12-s − 1.18·13-s + 0.675·14-s − 2.14·15-s + 1.62·16-s + 0.845·17-s − 4.37·18-s − 0.400·19-s + 2.54·20-s + 0.701·21-s − 0.539·22-s + 0.236·23-s + 3.97·24-s + 0.340·25-s + 2.11·26-s − 2.68·27-s − 0.830·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 5.05T + 8T^{2} \) |
| 3 | \( 1 + 9.64T + 27T^{2} \) |
| 5 | \( 1 - 12.9T + 125T^{2} \) |
| 13 | \( 1 + 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 26.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 79.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 593.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 49.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 546.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 809.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 85.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 750.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 451.T + 9.12e5T^{2} \) |
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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
−12.92637504752637039576106597423, −11.88337491923734153914492227989, −10.88071735527003557479835441964, −9.967921623863442184276883178611, −9.453554465884656392953467879988, −7.41000213023357838781104720931, −6.44833032670455203553731329261, −5.42217514794295291291569259983, −1.67315106252138622706386683089, 0,
1.67315106252138622706386683089, 5.42217514794295291291569259983, 6.44833032670455203553731329261, 7.41000213023357838781104720931, 9.453554465884656392953467879988, 9.967921623863442184276883178611, 10.88071735527003557479835441964, 11.88337491923734153914492227989, 12.92637504752637039576106597423
