| L(s) = 1 | − 5-s + 11-s − 5·13-s − 6·17-s + 2·19-s + 3·23-s + 25-s + 9·29-s − 4·31-s + 2·37-s − 9·41-s − 11·43-s + 3·47-s − 3·53-s − 55-s − 8·61-s + 5·65-s − 8·67-s + 6·71-s − 14·73-s + 10·79-s − 12·83-s + 6·85-s − 18·89-s − 2·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 1.38·13-s − 1.45·17-s + 0.458·19-s + 0.625·23-s + 1/5·25-s + 1.67·29-s − 0.718·31-s + 0.328·37-s − 1.40·41-s − 1.67·43-s + 0.437·47-s − 0.412·53-s − 0.134·55-s − 1.02·61-s + 0.620·65-s − 0.977·67-s + 0.712·71-s − 1.63·73-s + 1.12·79-s − 1.31·83-s + 0.650·85-s − 1.90·89-s − 0.205·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09801244821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09801244821\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.39892354346440, −11.98827774825769, −11.60764023355675, −11.18808892513182, −10.67966499116554, −10.13501523421513, −9.863594560739855, −9.153347148022973, −8.944374801917010, −8.301140795525361, −8.002551848855054, −7.290889727082129, −6.859701218443916, −6.751462101501533, −6.015283871323810, −5.348431835769152, −4.882623102761952, −4.516807472743759, −4.109006851841282, −3.247882160927404, −2.955866009547587, −2.365846866400327, −1.695605731479284, −1.120263674001800, −0.08229483970741084,
0.08229483970741084, 1.120263674001800, 1.695605731479284, 2.365846866400327, 2.955866009547587, 3.247882160927404, 4.109006851841282, 4.516807472743759, 4.882623102761952, 5.348431835769152, 6.015283871323810, 6.751462101501533, 6.859701218443916, 7.290889727082129, 8.002551848855054, 8.301140795525361, 8.944374801917010, 9.153347148022973, 9.863594560739855, 10.13501523421513, 10.67966499116554, 11.18808892513182, 11.60764023355675, 11.98827774825769, 12.39892354346440
