L(s) = 1 | + 2·3-s + 7-s + 9-s + 4·13-s + 6·17-s − 2·19-s + 2·21-s − 5·25-s − 4·27-s + 6·29-s − 4·31-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s − 4·57-s + 6·59-s − 8·61-s + 63-s + 4·67-s + 2·73-s − 10·75-s + 8·79-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.10·13-s + 1.45·17-s − 0.458·19-s + 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s − 0.529·57-s + 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.234·73-s − 1.15·75-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.101530499\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.101530499\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−11.06714116909225412617528172908, −10.04491938039307927961472194001, −9.195409247501138715663586603039, −8.222985047096640302551257734717, −7.87021772021814993834479658639, −6.47781377385923123584313505983, −5.36663802373794330843690451326, −3.93522119575008455702519349301, −3.07302685938302839530976846531, −1.64950502134773404742708641242,
1.64950502134773404742708641242, 3.07302685938302839530976846531, 3.93522119575008455702519349301, 5.36663802373794330843690451326, 6.47781377385923123584313505983, 7.87021772021814993834479658639, 8.222985047096640302551257734717, 9.195409247501138715663586603039, 10.04491938039307927961472194001, 11.06714116909225412617528172908