| L(s) = 1 | + 3.35·3-s − 5-s + 7-s + 8.27·9-s − 1.49·11-s − 5.92·13-s − 3.35·15-s − 7.21·17-s − 3.40·19-s + 3.35·21-s − 23-s + 25-s + 17.7·27-s − 6.23·29-s − 5.02·31-s − 5.02·33-s − 35-s − 8.37·37-s − 19.9·39-s − 8.37·41-s − 2.09·43-s − 8.27·45-s + 1.54·47-s + 49-s − 24.2·51-s + 6.30·53-s + 1.49·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 0.447·5-s + 0.377·7-s + 2.75·9-s − 0.451·11-s − 1.64·13-s − 0.867·15-s − 1.75·17-s − 0.780·19-s + 0.732·21-s − 0.208·23-s + 0.200·25-s + 3.40·27-s − 1.15·29-s − 0.903·31-s − 0.875·33-s − 0.169·35-s − 1.37·37-s − 3.18·39-s − 1.30·41-s − 0.319·43-s − 1.23·45-s + 0.225·47-s + 0.142·49-s − 3.39·51-s + 0.866·53-s + 0.201·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3.35T + 3T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 + 5.92T + 13T^{2} \) |
| 17 | \( 1 + 7.21T + 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 + 2.09T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 - 6.30T + 53T^{2} \) |
| 59 | \( 1 - 9.08T + 59T^{2} \) |
| 61 | \( 1 + 6.19T + 61T^{2} \) |
| 67 | \( 1 - 2.26T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 + 0.788T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 - 1.61T + 83T^{2} \) |
| 89 | \( 1 - 9.09T + 89T^{2} \) |
| 97 | \( 1 - 0.113T + 97T^{2} \) |
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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
−7.70810018949566946069274406942, −7.20232950242328205821804793331, −6.72950522337180911224775261435, −5.22021432906901742806224409044, −4.54969672973573239251443789008, −3.92784035172686778569982021469, −3.13472544770486194456189960961, −2.12108150736476709340992392943, −2.00991269766704716432604016025, 0,
2.00991269766704716432604016025, 2.12108150736476709340992392943, 3.13472544770486194456189960961, 3.92784035172686778569982021469, 4.54969672973573239251443789008, 5.22021432906901742806224409044, 6.72950522337180911224775261435, 7.20232950242328205821804793331, 7.70810018949566946069274406942
