| L(s) = 1 | − 5-s + 4·7-s − 11-s + 6·17-s − 4·19-s + 4·23-s + 25-s + 10·29-s + 6·31-s − 4·35-s + 2·37-s + 4·41-s + 10·43-s − 8·47-s + 9·49-s − 4·53-s + 55-s − 12·59-s − 6·61-s − 2·67-s − 8·71-s + 6·73-s − 4·77-s − 10·79-s + 8·83-s − 6·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 0.301·11-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.85·29-s + 1.07·31-s − 0.676·35-s + 0.328·37-s + 0.624·41-s + 1.52·43-s − 1.16·47-s + 9/7·49-s − 0.549·53-s + 0.134·55-s − 1.56·59-s − 0.768·61-s − 0.244·67-s − 0.949·71-s + 0.702·73-s − 0.455·77-s − 1.12·79-s + 0.878·83-s − 0.650·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.853879323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.853879323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.46388406192770, −12.14410594024199, −11.65292183695381, −11.29272102041829, −10.63358096843378, −10.57218284181633, −9.966530103144838, −9.326415791905895, −8.821569161910267, −8.353573856731696, −7.866459723484924, −7.781839616119433, −7.167542612439213, −6.474748747964494, −6.066482112009529, −5.484753637181770, −4.854347027401684, −4.567792300213925, −4.263973630974660, −3.351388148029986, −2.916526425308495, −2.406397056526144, −1.579656491185729, −1.139191661922732, −0.5691008346497892,
0.5691008346497892, 1.139191661922732, 1.579656491185729, 2.406397056526144, 2.916526425308495, 3.351388148029986, 4.263973630974660, 4.567792300213925, 4.854347027401684, 5.484753637181770, 6.066482112009529, 6.474748747964494, 7.167542612439213, 7.781839616119433, 7.866459723484924, 8.353573856731696, 8.821569161910267, 9.326415791905895, 9.966530103144838, 10.57218284181633, 10.63358096843378, 11.29272102041829, 11.65292183695381, 12.14410594024199, 12.46388406192770
