| L(s) = 1 | + 96.2·2-s − 729·3-s + 1.07e3·4-s + 2.66e4·5-s − 7.01e4·6-s + 2.74e5·7-s − 6.85e5·8-s + 5.31e5·9-s + 2.56e6·10-s + 8.00e5·11-s − 7.82e5·12-s − 3.00e6·13-s + 2.64e7·14-s − 1.93e7·15-s − 7.47e7·16-s − 1.21e8·17-s + 5.11e7·18-s + 4.02e8·19-s + 2.85e7·20-s − 2.00e8·21-s + 7.70e7·22-s − 6.66e8·23-s + 4.99e8·24-s − 5.12e8·25-s − 2.88e8·26-s − 3.87e8·27-s + 2.95e8·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 0.131·4-s + 0.761·5-s − 0.614·6-s + 0.882·7-s − 0.924·8-s + 0.333·9-s + 0.810·10-s + 0.136·11-s − 0.0756·12-s − 0.172·13-s + 0.938·14-s − 0.439·15-s − 1.11·16-s − 1.21·17-s + 0.354·18-s + 1.96·19-s + 0.0998·20-s − 0.509·21-s + 0.144·22-s − 0.938·23-s + 0.533·24-s − 0.419·25-s − 0.183·26-s − 0.192·27-s + 0.115·28-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 | \( 1 - 96.2T + 8.19e3T^{2} \) |
| 5 | \( 1 - 2.66e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 2.74e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 8.00e5T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.00e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.21e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 4.02e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.66e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.58e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.88e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.84e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 1.94e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.71e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 4.88e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.01e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 4.55e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 6.52e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.59e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.70e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 8.10e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 1.15e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.23e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 9.65e12T + 6.73e25T^{2} \) |
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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
−9.877250103112043040850999177467, −9.017229851569019903035422339563, −7.62200152165571635814165837881, −6.35853635082896537303872157571, −5.51642921700244537968372438701, −4.84200535841653086658012816622, −3.84799859340455635386296802422, −2.46907024749161627140156158027, −1.37622936819117273897365478523, 0,
1.37622936819117273897365478523, 2.46907024749161627140156158027, 3.84799859340455635386296802422, 4.84200535841653086658012816622, 5.51642921700244537968372438701, 6.35853635082896537303872157571, 7.62200152165571635814165837881, 9.017229851569019903035422339563, 9.877250103112043040850999177467
