| L(s) = 1 | − 3.37·3-s − 2.37·7-s + 8.37·9-s + 4.37·11-s − 13-s + 4.37·17-s + 4·19-s + 8·21-s − 1.37·23-s − 18.1·27-s + 5.74·29-s + 2.37·31-s − 14.7·33-s + 10·37-s + 3.37·39-s − 2.74·41-s + 9.37·43-s + 10.3·47-s − 1.37·49-s − 14.7·51-s + 0.255·53-s − 13.4·57-s − 4.37·59-s − 9.74·61-s − 19.8·63-s + 9.11·67-s + 4.62·69-s + ⋯ |
| L(s) = 1 | − 1.94·3-s − 0.896·7-s + 2.79·9-s + 1.31·11-s − 0.277·13-s + 1.06·17-s + 0.917·19-s + 1.74·21-s − 0.286·23-s − 3.48·27-s + 1.06·29-s + 0.426·31-s − 2.56·33-s + 1.64·37-s + 0.539·39-s − 0.428·41-s + 1.42·43-s + 1.51·47-s − 0.196·49-s − 2.06·51-s + 0.0350·53-s − 1.78·57-s − 0.569·59-s − 1.24·61-s − 2.50·63-s + 1.11·67-s + 0.557·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031330805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031330805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 5.74T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 9.37T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 0.255T + 53T^{2} \) |
| 59 | \( 1 + 4.37T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 4.74T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−7.936077004975233905380711406104, −7.18648145375285974894894415328, −6.62001296454795229437312924004, −5.97373191330450895565287370464, −5.59061052065453386592362525800, −4.54821075851884993665509231058, −4.03507581858245407103073891406, −2.95467962036418116007296994271, −1.35551910262819133187130433745, −0.69832267305548252595902574660,
0.69832267305548252595902574660, 1.35551910262819133187130433745, 2.95467962036418116007296994271, 4.03507581858245407103073891406, 4.54821075851884993665509231058, 5.59061052065453386592362525800, 5.97373191330450895565287370464, 6.62001296454795229437312924004, 7.18648145375285974894894415328, 7.936077004975233905380711406104
