| L(s) = 1 | + 4.33·3-s + 25.2·7-s − 8.22·9-s + 48.7·11-s + 10.6·13-s − 49.3·17-s − 132.·19-s + 109.·21-s − 135.·23-s − 152.·27-s − 175.·29-s − 207.·31-s + 211.·33-s − 195.·37-s + 46.3·39-s − 102.·41-s − 413.·43-s − 159.·47-s + 292.·49-s − 213.·51-s − 477.·53-s − 573.·57-s + 249.·59-s + 771.·61-s − 207.·63-s − 8.51·67-s − 584.·69-s + ⋯ |
| L(s) = 1 | + 0.833·3-s + 1.36·7-s − 0.304·9-s + 1.33·11-s + 0.228·13-s − 0.704·17-s − 1.59·19-s + 1.13·21-s − 1.22·23-s − 1.08·27-s − 1.12·29-s − 1.19·31-s + 1.11·33-s − 0.866·37-s + 0.190·39-s − 0.388·41-s − 1.46·43-s − 0.495·47-s + 0.853·49-s − 0.587·51-s − 1.23·53-s − 1.33·57-s + 0.550·59-s + 1.61·61-s − 0.414·63-s − 0.0155·67-s − 1.02·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4.33T + 27T^{2} \) |
| 7 | \( 1 - 25.2T + 343T^{2} \) |
| 11 | \( 1 - 48.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 49.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 132.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 477.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 771.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 8.51T + 3.00e5T^{2} \) |
| 71 | \( 1 + 254.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 207.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 299.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 939.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 316.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 172.T + 9.12e5T^{2} \) |
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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
−8.509684745866902776352143149649, −7.900533419096927851101208306429, −6.89463453670940938012046915524, −6.09845827291948360912299845385, −5.08298991853171684812830690424, −4.10731879428554441447792066667, −3.56010247656525214308841880261, −1.99977218290834871515412230292, −1.78424475190675052266985917472, 0,
1.78424475190675052266985917472, 1.99977218290834871515412230292, 3.56010247656525214308841880261, 4.10731879428554441447792066667, 5.08298991853171684812830690424, 6.09845827291948360912299845385, 6.89463453670940938012046915524, 7.900533419096927851101208306429, 8.509684745866902776352143149649
