L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 2·7-s − 3·8-s + 9-s + 12-s − 2·13-s − 2·14-s − 16-s + 18-s + 2·19-s + 2·21-s + 3·24-s − 5·25-s − 2·26-s − 27-s + 2·28-s + 10·29-s + 4·31-s + 5·32-s − 36-s − 2·37-s + 2·38-s + 2·39-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.436·21-s + 0.612·24-s − 25-s − 0.392·26-s − 0.192·27-s + 0.377·28-s + 1.85·29-s + 0.718·31-s + 0.883·32-s − 1/6·36-s − 0.328·37-s + 0.324·38-s + 0.320·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.33342694214820, −12.86734967593731, −12.38153000186621, −11.96231866557533, −11.81871026815123, −11.06608309237848, −10.41885461314047, −10.06206874092584, −9.584909653999330, −9.232381787001287, −8.646285579459074, −7.924698782182838, −7.704202226338026, −6.698591443629400, −6.557409848039354, −6.006052818562312, −5.512401341954062, −4.889726674638432, −4.618263805031860, −4.029540415038984, −3.371305148124905, −2.971537004407954, −2.355414044904951, −1.411853713953491, −0.6430154609368470, 0,
0.6430154609368470, 1.411853713953491, 2.355414044904951, 2.971537004407954, 3.371305148124905, 4.029540415038984, 4.618263805031860, 4.889726674638432, 5.512401341954062, 6.006052818562312, 6.557409848039354, 6.698591443629400, 7.704202226338026, 7.924698782182838, 8.646285579459074, 9.232381787001287, 9.584909653999330, 10.06206874092584, 10.41885461314047, 11.06608309237848, 11.81871026815123, 11.96231866557533, 12.38153000186621, 12.86734967593731, 13.33342694214820