| L(s) = 1 | + 27·3-s − 335.·5-s + 710.·7-s + 729·9-s + 2.89e3·11-s − 1.01e4·13-s − 9.05e3·15-s − 1.42e4·17-s + 3.31e4·19-s + 1.91e4·21-s + 7.53e4·23-s + 3.42e4·25-s + 1.96e4·27-s − 1.50e5·29-s − 5.51e4·31-s + 7.80e4·33-s − 2.38e5·35-s − 5.10e5·37-s − 2.73e5·39-s + 6.04e5·41-s − 4.79e5·43-s − 2.44e5·45-s − 1.59e5·47-s − 3.18e5·49-s − 3.84e5·51-s − 3.19e5·53-s − 9.69e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.19·5-s + 0.782·7-s + 0.333·9-s + 0.655·11-s − 1.27·13-s − 0.692·15-s − 0.703·17-s + 1.11·19-s + 0.451·21-s + 1.29·23-s + 0.438·25-s + 0.192·27-s − 1.14·29-s − 0.332·31-s + 0.378·33-s − 0.938·35-s − 1.65·37-s − 0.738·39-s + 1.37·41-s − 0.920·43-s − 0.399·45-s − 0.223·47-s − 0.387·49-s − 0.405·51-s − 0.294·53-s − 0.785·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 + 335.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 710.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.89e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.42e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.31e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.53e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.51e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.10e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.79e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.59e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.19e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.81e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.69e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.26e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.78e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.58e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.01e7T + 8.07e13T^{2} \) |
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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
−10.98616858538438537170612415414, −9.561474851904512813375408342583, −8.689713727279348893369282437203, −7.60486971419363210483031050716, −7.08427718477604612639810458046, −5.14260172369309533841908137007, −4.18241482488771285629178448986, −3.04323680147922211300873316915, −1.55427171671921840424117343047, 0,
1.55427171671921840424117343047, 3.04323680147922211300873316915, 4.18241482488771285629178448986, 5.14260172369309533841908137007, 7.08427718477604612639810458046, 7.60486971419363210483031050716, 8.689713727279348893369282437203, 9.561474851904512813375408342583, 10.98616858538438537170612415414
