| L(s) = 1 | − 16·2-s + 85.7·3-s + 256·4-s − 1.78e3·5-s − 1.37e3·6-s + 751.·7-s − 4.09e3·8-s − 1.23e4·9-s + 2.85e4·10-s − 5.85e4·11-s + 2.19e4·12-s − 1.20e4·14-s − 1.53e5·15-s + 6.55e4·16-s + 2.34e5·17-s + 1.97e5·18-s + 1.03e5·19-s − 4.57e5·20-s + 6.44e4·21-s + 9.36e5·22-s + 1.08e6·23-s − 3.51e5·24-s + 1.24e6·25-s − 2.74e6·27-s + 1.92e5·28-s + 3.17e6·29-s + 2.45e6·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.611·3-s + 0.5·4-s − 1.27·5-s − 0.432·6-s + 0.118·7-s − 0.353·8-s − 0.626·9-s + 0.904·10-s − 1.20·11-s + 0.305·12-s − 0.0837·14-s − 0.781·15-s + 0.250·16-s + 0.681·17-s + 0.442·18-s + 0.181·19-s − 0.639·20-s + 0.0723·21-s + 0.852·22-s + 0.810·23-s − 0.216·24-s + 0.635·25-s − 0.994·27-s + 0.0591·28-s + 0.833·29-s + 0.552·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 85.7T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.78e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 751.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.85e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.34e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.03e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.08e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.17e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.17e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.01e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.56e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.12e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.44e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.77e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 6.05e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.62e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.15e8T + 7.60e17T^{2} \) |
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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
−9.402831824429296316092501166065, −8.226532260854490607559335589146, −8.064299139953062536507748264228, −7.13097420311181749944085208068, −5.73250663911486460876144109823, −4.49436849782083243751187883724, −3.18435381333855627256655385649, −2.59811099000145756975991788452, −0.963690754446389215572336137258, 0,
0.963690754446389215572336137258, 2.59811099000145756975991788452, 3.18435381333855627256655385649, 4.49436849782083243751187883724, 5.73250663911486460876144109823, 7.13097420311181749944085208068, 8.064299139953062536507748264228, 8.226532260854490607559335589146, 9.402831824429296316092501166065
