L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 5·11-s + 14-s + 16-s − 2·17-s − 20-s − 5·22-s + 2·23-s + 25-s − 28-s − 10·29-s + 2·31-s − 32-s + 2·34-s + 35-s − 4·37-s + 40-s + 5·41-s + 4·43-s + 5·44-s − 2·46-s − 4·47-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 + 1.50·11-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.223·20-s − 1.06·22-s + 0.417·23-s + 1/5·25-s − 0.188·28-s − 1.85·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 0.657·37-s + 0.158·40-s + 0.780·41-s + 0.609·43-s + 0.753·44-s − 0.294·46-s − 0.583·47-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
\(L(1)\) |
\(\approx\) |
\(0.6402039187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6402039187\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−12.73035437295922, −12.42817765257322, −11.98598873457014, −11.31481303864058, −11.19342183615773, −10.72580030627670, −10.01374575953871, −9.533621616773966, −9.177360861750109, −8.885836070229752, −8.267716090055854, −7.735359733357666, −7.284710807785129, −6.693136660437872, −6.528821787482697, −5.798268969220084, −5.355993955266248, −4.503880869427381, −4.051013546422049, −3.605557079417278, −2.988800294756903, −2.374692843368804, −1.549918833393030, −1.234174189538865, −0.2588192866674218,
0.2588192866674218, 1.234174189538865, 1.549918833393030, 2.374692843368804, 2.988800294756903, 3.605557079417278, 4.051013546422049, 4.503880869427381, 5.355993955266248, 5.798268969220084, 6.528821787482697, 6.693136660437872, 7.284710807785129, 7.735359733357666, 8.267716090055854, 8.885836070229752, 9.177360861750109, 9.533621616773966, 10.01374575953871, 10.72580030627670, 11.19342183615773, 11.31481303864058, 11.98598873457014, 12.42817765257322, 12.73035437295922