| L(s) = 1 | + 5·3-s − 10·7-s − 2·9-s − 15·11-s − 8·13-s + 21·17-s + 105·19-s − 50·21-s − 10·23-s − 145·27-s − 20·29-s − 230·31-s − 75·33-s + 54·37-s − 40·39-s − 195·41-s − 300·43-s − 480·47-s − 243·49-s + 105·51-s − 322·53-s + 525·57-s + 560·59-s − 730·61-s + 20·63-s + 255·67-s − 50·69-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.539·7-s − 0.0740·9-s − 0.411·11-s − 0.170·13-s + 0.299·17-s + 1.26·19-s − 0.519·21-s − 0.0906·23-s − 1.03·27-s − 0.128·29-s − 1.33·31-s − 0.395·33-s + 0.239·37-s − 0.164·39-s − 0.742·41-s − 1.06·43-s − 1.48·47-s − 0.708·49-s + 0.288·51-s − 0.834·53-s + 1.21·57-s + 1.23·59-s − 1.53·61-s + 0.0399·63-s + 0.464·67-s − 0.0872·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 15 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 105 T + p^{3} T^{2} \) |
| 23 | \( 1 + 10 T + p^{3} T^{2} \) |
| 29 | \( 1 + 20 T + p^{3} T^{2} \) |
| 31 | \( 1 + 230 T + p^{3} T^{2} \) |
| 37 | \( 1 - 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 195 T + p^{3} T^{2} \) |
| 43 | \( 1 + 300 T + p^{3} T^{2} \) |
| 47 | \( 1 + 480 T + p^{3} T^{2} \) |
| 53 | \( 1 + 322 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 730 T + p^{3} T^{2} \) |
| 67 | \( 1 - 255 T + p^{3} T^{2} \) |
| 71 | \( 1 + 40 T + p^{3} T^{2} \) |
| 73 | \( 1 + 317 T + p^{3} T^{2} \) |
| 79 | \( 1 + 830 T + p^{3} T^{2} \) |
| 83 | \( 1 - 75 T + p^{3} T^{2} \) |
| 89 | \( 1 + 705 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1434 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.509445084894415534713261054788, −8.585373531331826151510281065990, −7.82968729056144618947962507145, −7.05207900476238995866176614092, −5.88447893607575885898331071031, −4.96123395463911190326894433166, −3.51924056057210916830799806023, −2.99945737284568456882594484046, −1.73285853205686140119700258745, 0,
1.73285853205686140119700258745, 2.99945737284568456882594484046, 3.51924056057210916830799806023, 4.96123395463911190326894433166, 5.88447893607575885898331071031, 7.05207900476238995866176614092, 7.82968729056144618947962507145, 8.585373531331826151510281065990, 9.509445084894415534713261054788
