L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 13-s + 14-s + 16-s + 4·17-s − 20-s + 4·23-s + 25-s − 26-s − 28-s + 6·29-s + 3·31-s − 32-s − 4·34-s + 35-s − 4·37-s + 40-s − 9·41-s − 9·43-s − 4·46-s − 9·47-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s + 1.11·29-s + 0.538·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s − 0.657·37-s + 0.158·40-s − 1.40·41-s − 1.37·43-s − 0.589·46-s − 1.31·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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
−13.12340698060432, −12.69019685362922, −11.95939622689021, −11.85337587467594, −11.37593993637756, −10.73513011234860, −10.29434833480077, −9.864153525546977, −9.636018868779061, −8.663656740331857, −8.558796289002450, −8.186967377152695, −7.469693655099151, −7.016864218084233, −6.643241314431904, −6.169808430012382, −5.406504515885808, −5.050907083706767, −4.423588886077101, −3.595161135163160, −3.300436163041872, −2.795742973104316, −2.006449757565504, −1.314695322540712, −0.7787997172010541, 0,
0.7787997172010541, 1.314695322540712, 2.006449757565504, 2.795742973104316, 3.300436163041872, 3.595161135163160, 4.423588886077101, 5.050907083706767, 5.406504515885808, 6.169808430012382, 6.643241314431904, 7.016864218084233, 7.469693655099151, 8.186967377152695, 8.558796289002450, 8.663656740331857, 9.636018868779061, 9.864153525546977, 10.29434833480077, 10.73513011234860, 11.37593993637756, 11.85337587467594, 11.95939622689021, 12.69019685362922, 13.12340698060432