| L(s) = 1 | − 12i·3-s + 88i·7-s + 99·9-s + 540·11-s − 418i·13-s − 594i·17-s − 836·19-s + 1.05e3·21-s − 4.10e3i·23-s − 4.10e3i·27-s + 594·29-s + 4.25e3·31-s − 6.48e3i·33-s + 298i·37-s − 5.01e3·39-s + ⋯ |
| L(s) = 1 | − 0.769i·3-s + 0.678i·7-s + 0.407·9-s + 1.34·11-s − 0.685i·13-s − 0.498i·17-s − 0.531·19-s + 0.522·21-s − 1.61i·23-s − 1.08i·27-s + 0.131·29-s + 0.795·31-s − 1.03i·33-s + 0.0357i·37-s − 0.528·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.68475 - 1.04123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68475 - 1.04123i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 12iT - 243T^{2} \) |
| 7 | \( 1 - 88iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 540T + 1.61e5T^{2} \) |
| 13 | \( 1 + 418iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 594iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 836T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 594T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 298iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.21e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.94e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 7.66e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.18e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.75e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.77e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 2.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.22e5iT - 8.58e9T^{2} \) |
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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
−12.51396853960862174236686145094, −12.05755404258012186985969404174, −10.65898617984728040008810232315, −9.335093011529123433969775117690, −8.274844268100986123986863333172, −6.97346389887750239379591014397, −6.03975676589006917803936088875, −4.33341920730405106686174517848, −2.46587976791816301424791845958, −0.913882871596800214533784300681,
1.40565260639191609453305475098, 3.71666978379047021140590370427, 4.50685572201736864303206024550, 6.27092091519014015084841797241, 7.44555903297766330524587557750, 9.026152304595149776645052717057, 9.822502134501987913785763487523, 10.89708806219967913385458770068, 11.89661695338397288241979451445, 13.20815479895680098001564353237
