| L(s) = 1 | − 80.4·2-s − 729·3-s − 1.71e3·4-s − 1.90e4·5-s + 5.86e4·6-s − 2.13e5·7-s + 7.97e5·8-s + 5.31e5·9-s + 1.53e6·10-s + 8.17e6·11-s + 1.24e6·12-s + 1.28e7·13-s + 1.71e7·14-s + 1.39e7·15-s − 5.01e7·16-s − 1.73e8·17-s − 4.27e7·18-s − 1.17e8·19-s + 3.26e7·20-s + 1.55e8·21-s − 6.58e8·22-s − 4.16e8·23-s − 5.81e8·24-s − 8.56e8·25-s − 1.03e9·26-s − 3.87e8·27-s + 3.65e8·28-s + ⋯ |
| L(s) = 1 | − 0.889·2-s − 0.577·3-s − 0.208·4-s − 0.545·5-s + 0.513·6-s − 0.686·7-s + 1.07·8-s + 0.333·9-s + 0.485·10-s + 1.39·11-s + 0.120·12-s + 0.737·13-s + 0.610·14-s + 0.315·15-s − 0.747·16-s − 1.74·17-s − 0.296·18-s − 0.575·19-s + 0.114·20-s + 0.396·21-s − 1.23·22-s − 0.586·23-s − 0.620·24-s − 0.701·25-s − 0.655·26-s − 0.192·27-s + 0.143·28-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
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 80.4T + 8.19e3T^{2} \) |
| 5 | \( 1 + 1.90e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.13e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 8.17e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.28e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.73e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.17e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 4.16e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.55e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 3.96e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.91e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.39e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 5.38e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 9.05e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.89e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 1.44e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.53e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 8.31e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.24e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.36e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.42e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 1.71e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.14e13T + 6.73e25T^{2} \) |
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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
−9.598588532414572697380317754363, −9.039917601404044100344460956997, −8.003265059902291248013715605010, −6.82803886173078646719316468303, −6.06805212201412867198265967594, −4.36434699744165611721774488043, −3.87573150448617153272810517566, −1.97104879364643632688870604431, −0.814435619741079473914245416501, 0,
0.814435619741079473914245416501, 1.97104879364643632688870604431, 3.87573150448617153272810517566, 4.36434699744165611721774488043, 6.06805212201412867198265967594, 6.82803886173078646719316468303, 8.003265059902291248013715605010, 9.039917601404044100344460956997, 9.598588532414572697380317754363
