| L(s) = 1 | − 2.99·3-s + 9.67·5-s + 6.01·7-s − 18.0·9-s − 14.8·11-s + 13·13-s − 28.9·15-s + 71.8·17-s − 129.·19-s − 18.0·21-s + 99.1·23-s − 31.3·25-s + 134.·27-s + 121.·29-s − 22.4·31-s + 44.5·33-s + 58.2·35-s + 366.·37-s − 38.9·39-s − 470.·41-s − 361.·43-s − 174.·45-s − 131.·47-s − 306.·49-s − 215.·51-s − 525.·53-s − 144.·55-s + ⋯ |
| L(s) = 1 | − 0.576·3-s + 0.865·5-s + 0.325·7-s − 0.667·9-s − 0.408·11-s + 0.277·13-s − 0.498·15-s + 1.02·17-s − 1.56·19-s − 0.187·21-s + 0.898·23-s − 0.250·25-s + 0.961·27-s + 0.775·29-s − 0.129·31-s + 0.235·33-s + 0.281·35-s + 1.62·37-s − 0.159·39-s − 1.79·41-s − 1.28·43-s − 0.578·45-s − 0.407·47-s − 0.894·49-s − 0.590·51-s − 1.36·53-s − 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 2.99T + 27T^{2} \) |
| 5 | \( 1 - 9.67T + 125T^{2} \) |
| 7 | \( 1 - 6.01T + 343T^{2} \) |
| 11 | \( 1 + 14.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 71.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 22.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 366.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 470.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 361.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 131.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 305.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 456.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 719.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 52.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 922.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 390.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−8.469144471644995446759717291853, −8.034965562867695303570591773004, −6.67837885612082065437813358097, −6.18956258537665054445415857718, −5.33241282796201892164915208954, −4.75035450450050779588792075709, −3.37764501651926127097994962662, −2.37547263668618028578780269358, −1.29749168091260277190736784499, 0,
1.29749168091260277190736784499, 2.37547263668618028578780269358, 3.37764501651926127097994962662, 4.75035450450050779588792075709, 5.33241282796201892164915208954, 6.18956258537665054445415857718, 6.67837885612082065437813358097, 8.034965562867695303570591773004, 8.469144471644995446759717291853
