| L(s) = 1 | + 442.·2-s − 6.56e3·3-s + 6.47e4·4-s − 3.90e5·5-s − 2.90e6·6-s + 2.47e7·7-s − 2.93e7·8-s + 4.30e7·9-s − 1.72e8·10-s − 7.66e6·11-s − 4.25e8·12-s − 3.40e9·13-s + 1.09e10·14-s + 2.56e9·15-s − 2.14e10·16-s − 5.35e10·17-s + 1.90e10·18-s − 1.29e11·19-s − 2.53e10·20-s − 1.62e11·21-s − 3.39e9·22-s + 2.11e11·23-s + 1.92e11·24-s + 1.52e11·25-s − 1.50e12·26-s − 2.82e11·27-s + 1.60e12·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.494·4-s − 0.447·5-s − 0.705·6-s + 1.62·7-s − 0.618·8-s + 0.333·9-s − 0.546·10-s − 0.0107·11-s − 0.285·12-s − 1.15·13-s + 1.98·14-s + 0.258·15-s − 1.24·16-s − 1.86·17-s + 0.407·18-s − 1.75·19-s − 0.221·20-s − 0.938·21-s − 0.0131·22-s + 0.562·23-s + 0.356·24-s + 0.200·25-s − 1.41·26-s − 0.192·27-s + 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 2 | \( 1 - 442.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.47e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.66e6T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.40e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 5.35e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.29e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.11e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 4.13e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 5.31e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.85e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.57e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 7.31e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.22e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 2.74e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 6.31e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.14e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.08e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 1.25e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 7.71e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.63e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.45e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 2.61e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 9.64e15T + 5.95e33T^{2} \) |
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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
−14.60428419655224266854882885979, −13.08752123880097251436015134225, −11.87349428160774636125167395137, −10.92502758303279075185016564186, −8.543263766370436674950230513915, −6.72385735924476278848349701774, −4.94327793912502887071678982299, −4.38929770231596239220908569500, −2.19054759808334207002216786539, 0,
2.19054759808334207002216786539, 4.38929770231596239220908569500, 4.94327793912502887071678982299, 6.72385735924476278848349701774, 8.543263766370436674950230513915, 10.92502758303279075185016564186, 11.87349428160774636125167395137, 13.08752123880097251436015134225, 14.60428419655224266854882885979
