L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 4.89·7-s + 8-s + 9-s + 12-s + 2.89·13-s − 4.89·14-s + 16-s + 17-s + 18-s + 4·19-s − 4.89·21-s + 4·23-s + 24-s + 2.89·26-s + 27-s − 4.89·28-s + 6·29-s + 4·31-s + 32-s + 34-s + 36-s − 6·37-s + 4·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.85·7-s + 0.353·8-s + 0.333·9-s + 0.288·12-s + 0.804·13-s − 1.30·14-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 1.06·21-s + 0.834·23-s + 0.204·24-s + 0.568·26-s + 0.192·27-s − 0.925·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.166·36-s − 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.145442388\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.145442388\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 43 | \( 1 - 0.898T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 7.79T + 61T^{2} \) |
| 67 | \( 1 - 8.89T + 67T^{2} \) |
| 71 | \( 1 - 0.898T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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
−9.082550147459862918676075951706, −8.062299306457441686802596270243, −7.18828732564724250752809734899, −6.50667932303110474761670697883, −5.94152664604761259414837900025, −4.90474911446282460543269684986, −3.81978772612130592958545511318, −3.22895280552767533646838242355, −2.60829502822413148291791873420, −1.01371369103375821978881681616,
1.01371369103375821978881681616, 2.60829502822413148291791873420, 3.22895280552767533646838242355, 3.81978772612130592958545511318, 4.90474911446282460543269684986, 5.94152664604761259414837900025, 6.50667932303110474761670697883, 7.18828732564724250752809734899, 8.062299306457441686802596270243, 9.082550147459862918676075951706