| L(s) = 1 | − 4·13-s + 6·17-s + 2·19-s − 5·25-s − 6·29-s + 4·31-s − 2·37-s + 6·41-s − 8·43-s − 12·47-s + 6·53-s + 6·59-s + 8·61-s + 4·67-s − 2·73-s + 8·79-s + 6·83-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.10·13-s + 1.45·17-s + 0.458·19-s − 25-s − 1.11·29-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 0.824·53-s + 0.781·59-s + 1.02·61-s + 0.488·67-s − 0.234·73-s + 0.900·79-s + 0.658·83-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.35231965333849, −14.96976260059801, −14.29482422502816, −14.14204513626903, −13.17970777485948, −12.95530419283022, −12.16275059698363, −11.72653549176093, −11.42453006806420, −10.47349085550044, −9.941839656954204, −9.707432703860519, −9.040694804518053, −8.180875231063652, −7.801622845514357, −7.284696382845970, −6.625375357727742, −5.902618453702980, −5.254970731200679, −4.919080375737291, −3.897101947701589, −3.457023811465238, −2.625166654235116, −1.923349151904354, −1.030370765192138, 0,
1.030370765192138, 1.923349151904354, 2.625166654235116, 3.457023811465238, 3.897101947701589, 4.919080375737291, 5.254970731200679, 5.902618453702980, 6.625375357727742, 7.284696382845970, 7.801622845514357, 8.180875231063652, 9.040694804518053, 9.707432703860519, 9.941839656954204, 10.47349085550044, 11.42453006806420, 11.72653549176093, 12.16275059698363, 12.95530419283022, 13.17970777485948, 14.14204513626903, 14.29482422502816, 14.96976260059801, 15.35231965333849
