L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 6·11-s − 4·13-s − 15-s − 6·17-s + 4·19-s + 21-s + 23-s + 25-s − 27-s + 4·31-s − 6·33-s − 35-s − 10·37-s + 4·39-s + 6·41-s + 10·43-s + 45-s + 49-s + 6·51-s − 6·53-s + 6·55-s − 4·57-s + 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s − 1.04·33-s − 0.169·35-s − 1.64·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.809·55-s − 0.529·57-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.836721604\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.836721604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−14.66061341886615, −14.35798454727929, −13.77985434900626, −13.32381574180816, −12.61219213588123, −12.04557307796903, −11.93715966794245, −11.04348937301613, −10.81220554882557, −9.863542394250030, −9.642618046487272, −9.002804852720004, −8.713437947521884, −7.597762460229527, −7.136349981365030, −6.627587890736931, −6.184899596422231, −5.578547148611908, −4.782019831541931, −4.393504741706071, −3.672167011114094, −2.874278689075794, −2.109404549480808, −1.373135327837221, −0.5374141695602303,
0.5374141695602303, 1.373135327837221, 2.109404549480808, 2.874278689075794, 3.672167011114094, 4.393504741706071, 4.782019831541931, 5.578547148611908, 6.184899596422231, 6.627587890736931, 7.136349981365030, 7.597762460229527, 8.713437947521884, 9.002804852720004, 9.642618046487272, 9.863542394250030, 10.81220554882557, 11.04348937301613, 11.93715966794245, 12.04557307796903, 12.61219213588123, 13.32381574180816, 13.77985434900626, 14.35798454727929, 14.66061341886615