| L(s) = 1 | − 50.5·2-s + 729·3-s − 5.63e3·4-s + 3.74e4·5-s − 3.68e4·6-s + 1.31e5·7-s + 6.99e5·8-s + 5.31e5·9-s − 1.89e6·10-s + 4.83e6·11-s − 4.10e6·12-s + 7.84e6·13-s − 6.64e6·14-s + 2.73e7·15-s + 1.08e7·16-s + 1.27e8·17-s − 2.68e7·18-s + 1.07e8·19-s − 2.11e8·20-s + 9.58e7·21-s − 2.44e8·22-s + 4.53e8·23-s + 5.09e8·24-s + 1.83e8·25-s − 3.96e8·26-s + 3.87e8·27-s − 7.41e8·28-s + ⋯ |
| L(s) = 1 | − 0.558·2-s + 0.577·3-s − 0.688·4-s + 1.07·5-s − 0.322·6-s + 0.422·7-s + 0.942·8-s + 0.333·9-s − 0.599·10-s + 0.823·11-s − 0.397·12-s + 0.450·13-s − 0.235·14-s + 0.619·15-s + 0.161·16-s + 1.28·17-s − 0.186·18-s + 0.526·19-s − 0.738·20-s + 0.243·21-s − 0.459·22-s + 0.638·23-s + 0.544·24-s + 0.150·25-s − 0.251·26-s + 0.192·27-s − 0.290·28-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
| \(L(7)\) |
\(\approx\) |
\(3.346239833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.346239833\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 50.5T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.74e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 1.31e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 4.83e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 7.84e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.27e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.07e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 4.53e8T + 5.04e17T^{2} \) |
| 29 | \( 1 - 2.96e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 5.30e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.26e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 4.56e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.06e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 3.67e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.16e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.48e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 7.76e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 6.93e9T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.40e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.00e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.96e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.15e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 3.57e12T + 6.73e25T^{2} \) |
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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
−9.834703590146575928546565479863, −9.505726843500138730341415625976, −8.480742414037217758130211611893, −7.67711983650930485486527339018, −6.30286342758068724236473173181, −5.18232797015386100819961283305, −4.09570180108831347271206865327, −2.83890459352242005154179878005, −1.43306229773438646734449471651, −0.984801374974415170397633014926,
0.984801374974415170397633014926, 1.43306229773438646734449471651, 2.83890459352242005154179878005, 4.09570180108831347271206865327, 5.18232797015386100819961283305, 6.30286342758068724236473173181, 7.67711983650930485486527339018, 8.480742414037217758130211611893, 9.505726843500138730341415625976, 9.834703590146575928546565479863
