| L(s) = 1 | − 0.710·3-s − 4.59·7-s − 2.49·9-s − 3.91·11-s + 0.572·13-s + 0.232·17-s + 5.55·19-s + 3.26·21-s − 4.93·23-s + 3.90·27-s + 4.13·29-s + 3.49·31-s + 2.78·33-s + 5.41·37-s − 0.406·39-s + 10.4·41-s + 1.38·43-s + 0.920·47-s + 14.0·49-s − 0.165·51-s − 1.23·53-s − 3.94·57-s − 4.50·59-s − 11.6·61-s + 11.4·63-s − 2.95·67-s + 3.50·69-s + ⋯ |
| L(s) = 1 | − 0.410·3-s − 1.73·7-s − 0.831·9-s − 1.18·11-s + 0.158·13-s + 0.0564·17-s + 1.27·19-s + 0.711·21-s − 1.02·23-s + 0.751·27-s + 0.767·29-s + 0.627·31-s + 0.484·33-s + 0.890·37-s − 0.0651·39-s + 1.62·41-s + 0.211·43-s + 0.134·47-s + 2.01·49-s − 0.0231·51-s − 0.169·53-s − 0.522·57-s − 0.586·59-s − 1.49·61-s + 1.44·63-s − 0.360·67-s + 0.421·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.710T + 3T^{2} \) |
| 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 - 0.572T + 13T^{2} \) |
| 17 | \( 1 - 0.232T + 17T^{2} \) |
| 19 | \( 1 - 5.55T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 - 0.920T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 2.95T + 67T^{2} \) |
| 71 | \( 1 + 3.20T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 9.61T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
| 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
−7.38393878251803144718789991760, −6.30437583992008388904362606558, −6.11729005267561152662631617271, −5.43961903880285467753671718902, −4.63279265829320472340592230543, −3.64708999302737662731615609246, −2.90017306338988595008162935809, −2.53413735459352264717614806702, −0.875456247795245112319315984176, 0,
0.875456247795245112319315984176, 2.53413735459352264717614806702, 2.90017306338988595008162935809, 3.64708999302737662731615609246, 4.63279265829320472340592230543, 5.43961903880285467753671718902, 6.11729005267561152662631617271, 6.30437583992008388904362606558, 7.38393878251803144718789991760
