| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s − 6·11-s + 13-s − 15-s + 6·17-s + 2·19-s − 21-s + 23-s + 25-s − 5·27-s − 9·29-s + 31-s − 6·33-s + 35-s − 8·37-s + 39-s + 9·41-s + 2·43-s + 2·45-s + 3·47-s + 49-s + 6·51-s + 6·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.80·11-s + 0.277·13-s − 0.258·15-s + 1.45·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 0.179·31-s − 1.04·33-s + 0.169·35-s − 1.31·37-s + 0.160·39-s + 1.40·41-s + 0.304·43-s + 0.298·45-s + 0.437·47-s + 1/7·49-s + 0.840·51-s + 0.809·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.71887221088168, −14.20738617658315, −13.80096604515144, −13.15540555880635, −12.79063389424445, −12.31295121942469, −11.60882332590102, −11.15651725884875, −10.60439537504444, −10.07072468810598, −9.582151759808300, −8.907612954837759, −8.474310414016319, −7.778930606260591, −7.591312723248941, −7.066623745629562, −6.016327707872345, −5.508423865833691, −5.308513567908617, −4.332873853567846, −3.542353276179457, −3.200792621287571, −2.621611083153733, −1.935168020984895, −0.8204066279934325, 0,
0.8204066279934325, 1.935168020984895, 2.621611083153733, 3.200792621287571, 3.542353276179457, 4.332873853567846, 5.308513567908617, 5.508423865833691, 6.016327707872345, 7.066623745629562, 7.591312723248941, 7.778930606260591, 8.474310414016319, 8.907612954837759, 9.582151759808300, 10.07072468810598, 10.60439537504444, 11.15651725884875, 11.60882332590102, 12.31295121942469, 12.79063389424445, 13.15540555880635, 13.80096604515144, 14.20738617658315, 14.71887221088168
