L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 2·11-s + 12-s + 5·13-s + 14-s + 15-s + 16-s − 18-s + 20-s − 21-s − 2·22-s + 4·23-s − 24-s + 25-s − 5·26-s + 27-s − 28-s + 2·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s − 0.218·21-s − 0.426·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.272419333\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.272419333\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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
−12.56311495186337, −11.95058379518796, −11.49804036887330, −11.04905281307204, −10.60845734482810, −10.19460853107591, −9.665983637636126, −9.210547451333487, −8.943232612063193, −8.511917266677477, −8.082113801097384, −7.395772116909860, −7.126941003574244, −6.432584428084941, −6.157533134708668, −5.750117700857337, −4.977401039675919, −4.395404474681636, −3.828010358876215, −3.250451012564800, −2.959826378919227, −2.134518650902233, −1.758117867960729, −0.9609167810718557, −0.7009807000031671,
0.7009807000031671, 0.9609167810718557, 1.758117867960729, 2.134518650902233, 2.959826378919227, 3.250451012564800, 3.828010358876215, 4.395404474681636, 4.977401039675919, 5.750117700857337, 6.157533134708668, 6.432584428084941, 7.126941003574244, 7.395772116909860, 8.082113801097384, 8.511917266677477, 8.943232612063193, 9.210547451333487, 9.665983637636126, 10.19460853107591, 10.60845734482810, 11.04905281307204, 11.49804036887330, 11.95058379518796, 12.56311495186337