| L(s) = 1 | − 128·2-s − 2.50e3·3-s + 1.63e4·4-s − 1.54e5·5-s + 3.20e5·6-s − 2.09e6·8-s − 8.09e6·9-s + 1.98e7·10-s − 2.08e7·11-s − 4.09e7·12-s + 1.06e8·13-s + 3.86e8·15-s + 2.68e8·16-s − 2.34e9·17-s + 1.03e9·18-s + 1.94e9·19-s − 2.53e9·20-s + 2.66e9·22-s + 1.55e10·23-s + 5.24e9·24-s − 6.56e9·25-s − 1.36e10·26-s + 5.61e10·27-s − 4.01e10·29-s − 4.95e10·30-s − 9.58e10·31-s − 3.43e10·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.660·3-s + 0.5·4-s − 0.886·5-s + 0.466·6-s − 0.353·8-s − 0.564·9-s + 0.626·10-s − 0.322·11-s − 0.330·12-s + 0.471·13-s + 0.584·15-s + 0.250·16-s − 1.38·17-s + 0.399·18-s + 0.499·19-s − 0.443·20-s + 0.228·22-s + 0.949·23-s + 0.233·24-s − 0.214·25-s − 0.333·26-s + 1.03·27-s − 0.432·29-s − 0.413·30-s − 0.625·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 128T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.50e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.54e5T + 3.05e10T^{2} \) |
| 11 | \( 1 + 2.08e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.06e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.34e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.94e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.55e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 4.01e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 9.58e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.36e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.26e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 3.56e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 2.06e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 4.41e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 3.62e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.26e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 1.01e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.08e14T + 5.87e27T^{2} \) |
| 73 | \( 1 - 2.15e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.41e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.99e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 7.39e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 8.03e14T + 6.33e29T^{2} \) |
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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
−10.89154443146265034879749642660, −9.316998403325896251388205177111, −8.386686159897592541398041991610, −7.35015671686176769777863326254, −6.27668359732539413340841635127, −5.08557185430345931805857704978, −3.70296609677278217195543591770, −2.39328111416619906354905388769, −0.840612142932281467632883911001, 0,
0.840612142932281467632883911001, 2.39328111416619906354905388769, 3.70296609677278217195543591770, 5.08557185430345931805857704978, 6.27668359732539413340841635127, 7.35015671686176769777863326254, 8.386686159897592541398041991610, 9.316998403325896251388205177111, 10.89154443146265034879749642660
