| L(s) = 1 | − 147.·2-s − 729·3-s + 1.34e4·4-s + 1.77e4·5-s + 1.07e5·6-s − 5.61e5·7-s − 7.80e5·8-s + 5.31e5·9-s − 2.60e6·10-s + 4.98e6·11-s − 9.83e6·12-s + 1.18e7·13-s + 8.27e7·14-s − 1.29e7·15-s + 4.37e6·16-s + 4.20e7·17-s − 7.82e7·18-s − 2.49e6·19-s + 2.39e8·20-s + 4.09e8·21-s − 7.34e8·22-s + 6.03e8·23-s + 5.68e8·24-s − 9.06e8·25-s − 1.73e9·26-s − 3.87e8·27-s − 7.58e9·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.64·4-s + 0.507·5-s + 0.939·6-s − 1.80·7-s − 1.05·8-s + 0.333·9-s − 0.825·10-s + 0.848·11-s − 0.950·12-s + 0.678·13-s + 2.93·14-s − 0.292·15-s + 0.0652·16-s + 0.423·17-s − 0.542·18-s − 0.0121·19-s + 0.835·20-s + 1.04·21-s − 1.38·22-s + 0.850·23-s + 0.607·24-s − 0.742·25-s − 1.10·26-s − 0.192·27-s − 2.97·28-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
| \(L(7)\) |
\(\approx\) |
\(0.7006465788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7006465788\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 147.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 1.77e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 5.61e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 4.98e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.18e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 4.20e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.49e6T + 4.20e16T^{2} \) |
| 23 | \( 1 - 6.03e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.27e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 5.64e8T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.15e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.08e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 6.36e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 3.52e9T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.84e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 9.98e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.38e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.17e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 5.91e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 3.65e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 3.13e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 7.80e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 4.93e12T + 6.73e25T^{2} \) |
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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
−10.07204484349301340987691714743, −9.438180857548033473723423915315, −8.733976638882679079482601171154, −7.25579313886430084367835458661, −6.53049667129272263554599007977, −5.79614543698931308077697452056, −3.86178007908285315940785937620, −2.60040204711513486689202180037, −1.25891858917520058508238622230, −0.53573214434738985599639318258,
0.53573214434738985599639318258, 1.25891858917520058508238622230, 2.60040204711513486689202180037, 3.86178007908285315940785937620, 5.79614543698931308077697452056, 6.53049667129272263554599007977, 7.25579313886430084367835458661, 8.733976638882679079482601171154, 9.438180857548033473723423915315, 10.07204484349301340987691714743
