| L(s) = 1 | + 30.9·2-s − 729·3-s − 7.23e3·4-s − 9.68e3·5-s − 2.25e4·6-s − 5.22e4·7-s − 4.77e5·8-s + 5.31e5·9-s − 2.99e5·10-s + 8.65e6·11-s + 5.27e6·12-s − 9.70e6·13-s − 1.61e6·14-s + 7.06e6·15-s + 4.44e7·16-s − 2.53e7·17-s + 1.64e7·18-s + 7.70e7·19-s + 7.00e7·20-s + 3.81e7·21-s + 2.67e8·22-s + 5.41e7·23-s + 3.48e8·24-s − 1.12e9·25-s − 3.00e8·26-s − 3.87e8·27-s + 3.78e8·28-s + ⋯ |
| L(s) = 1 | + 0.341·2-s − 0.577·3-s − 0.883·4-s − 0.277·5-s − 0.197·6-s − 0.167·7-s − 0.643·8-s + 0.333·9-s − 0.0947·10-s + 1.47·11-s + 0.509·12-s − 0.557·13-s − 0.0574·14-s + 0.160·15-s + 0.662·16-s − 0.254·17-s + 0.113·18-s + 0.375·19-s + 0.244·20-s + 0.0969·21-s + 0.503·22-s + 0.0763·23-s + 0.371·24-s − 0.923·25-s − 0.190·26-s − 0.192·27-s + 0.148·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.001076659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001076659\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 - 30.9T + 8.19e3T^{2} \) |
| 5 | \( 1 + 9.68e3T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.22e4T + 9.68e10T^{2} \) |
| 11 | \( 1 - 8.65e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 9.70e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 2.53e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 7.70e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.41e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.17e7T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.81e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.22e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 4.38e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.92e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 6.56e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.94e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 4.89e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 4.70e9T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.35e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 4.40e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.66e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.61e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 7.16e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 7.32e12T + 6.73e25T^{2} \) |
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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
−10.22258432812441662962202240460, −9.401309690327407025992879670319, −8.488652476213928590128820274579, −7.14676229886517687725488075933, −6.15450751823762383747750728981, −5.09561456173349884793735552038, −4.18654067796879522634212886947, −3.34230274566234198056095420301, −1.59539779398710261443220915583, −0.43127019426374280988972244664,
0.43127019426374280988972244664, 1.59539779398710261443220915583, 3.34230274566234198056095420301, 4.18654067796879522634212886947, 5.09561456173349884793735552038, 6.15450751823762383747750728981, 7.14676229886517687725488075933, 8.488652476213928590128820274579, 9.401309690327407025992879670319, 10.22258432812441662962202240460
