| L(s) = 1 | + 2.46·5-s + 6.46·13-s + 5.92·17-s + 1.07·25-s − 1.53·29-s + 9.39·37-s + 10·41-s − 7·49-s − 14·53-s − 15.3·61-s + 15.9·65-s + 16.8·73-s + 14.6·85-s − 18.8·89-s + 18·97-s + 2·101-s − 14.3·109-s + 20.8·113-s + ⋯ |
| L(s) = 1 | + 1.10·5-s + 1.79·13-s + 1.43·17-s + 0.214·25-s − 0.285·29-s + 1.54·37-s + 1.56·41-s − 49-s − 1.92·53-s − 1.97·61-s + 1.97·65-s + 1.97·73-s + 1.58·85-s − 1.99·89-s + 1.82·97-s + 0.199·101-s − 1.37·109-s + 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.973954559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.973954559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 18.8T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
| 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
−8.077050173209106070326267514398, −7.70624430007089046212799027795, −6.47436308696754808961874685495, −6.04161228800115391918699617432, −5.55759572377750697368283758701, −4.55532150381668773045619297085, −3.61740212162844662839798720138, −2.87993906012818302723350816276, −1.74120521064781890205680362285, −1.03071923477985079780402782356,
1.03071923477985079780402782356, 1.74120521064781890205680362285, 2.87993906012818302723350816276, 3.61740212162844662839798720138, 4.55532150381668773045619297085, 5.55759572377750697368283758701, 6.04161228800115391918699617432, 6.47436308696754808961874685495, 7.70624430007089046212799027795, 8.077050173209106070326267514398
