| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s − 13-s + 14-s + 16-s − 6·17-s − 18-s − 4·19-s + 21-s + 24-s + 26-s − 27-s − 28-s + 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s + 10·37-s + 4·38-s + 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.648·38-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.53577972574953, −16.06042173879892, −15.46240217490684, −14.98258435215051, −14.37378789391304, −13.47768167753301, −13.00110603842114, −12.50639978362969, −11.78537174570469, −11.27142017810084, −10.72856954948314, −10.24505661118163, −9.548189673522124, −9.033544564490981, −8.408822302098008, −7.739885161079115, −7.001319631707097, −6.438020498655731, −6.083471804991526, −5.079527152899175, −4.453031973045118, −3.700649172011494, −2.602314594017692, −2.069974889439539, −0.8973555198840396, 0,
0.8973555198840396, 2.069974889439539, 2.602314594017692, 3.700649172011494, 4.453031973045118, 5.079527152899175, 6.083471804991526, 6.438020498655731, 7.001319631707097, 7.739885161079115, 8.408822302098008, 9.033544564490981, 9.548189673522124, 10.24505661118163, 10.72856954948314, 11.27142017810084, 11.78537174570469, 12.50639978362969, 13.00110603842114, 13.47768167753301, 14.37378789391304, 14.98258435215051, 15.46240217490684, 16.06042173879892, 16.53577972574953
