L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 15-s + 16-s − 2·17-s − 18-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 32-s − 4·33-s + 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.92894689250583, −12.51665218229175, −11.96835523845333, −11.68543554719052, −11.30488694412928, −10.71772072672434, −10.24445117388130, −9.838509031094479, −9.352268086809353, −9.125428691716929, −8.337260917514946, −8.062673517648231, −7.390847097285031, −6.868946400981919, −6.413228731749327, −6.197283031911989, −5.434820207559142, −5.078795793041125, −4.408931888368453, −3.571922255617752, −3.535267107987864, −2.438115551176012, −1.873420258469074, −1.484771732673329, −0.7422826382116539, 0,
0.7422826382116539, 1.484771732673329, 1.873420258469074, 2.438115551176012, 3.535267107987864, 3.571922255617752, 4.408931888368453, 5.078795793041125, 5.434820207559142, 6.197283031911989, 6.413228731749327, 6.868946400981919, 7.390847097285031, 8.062673517648231, 8.337260917514946, 9.125428691716929, 9.352268086809353, 9.838509031094479, 10.24445117388130, 10.71772072672434, 11.30488694412928, 11.68543554719052, 11.96835523845333, 12.51665218229175, 12.92894689250583