| L(s) = 1 | − 1.41·3-s + 1.41·7-s − 0.999·9-s + 0.732·11-s + 1.03·13-s + 0.378·17-s − 4.19·19-s − 2.00·21-s − 2.44·23-s + 5.65·27-s − 29-s + 0.196·31-s − 1.03·33-s + 0.378·37-s − 1.46·39-s − 5.46·41-s − 1.41·43-s + 8.38·47-s − 5·49-s − 0.535·51-s + 1.03·53-s + 5.93·57-s − 2·59-s − 2.53·61-s − 1.41·63-s − 1.69·67-s + 3.46·69-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 0.534·7-s − 0.333·9-s + 0.220·11-s + 0.287·13-s + 0.0919·17-s − 0.962·19-s − 0.436·21-s − 0.510·23-s + 1.08·27-s − 0.185·29-s + 0.0352·31-s − 0.180·33-s + 0.0622·37-s − 0.234·39-s − 0.853·41-s − 0.215·43-s + 1.22·47-s − 0.714·49-s − 0.0750·51-s + 0.142·53-s + 0.786·57-s − 0.260·59-s − 0.324·61-s − 0.178·63-s − 0.206·67-s + 0.417·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 - 0.378T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 31 | \( 1 - 0.196T + 31T^{2} \) |
| 37 | \( 1 - 0.378T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 2.53T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 + 8.92T + 71T^{2} \) |
| 73 | \( 1 + 0.656T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 0.535T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.420071575848007910972502904069, −7.66833359605819690013889104167, −6.67261145699874479486648339380, −6.09693617243412784458763011094, −5.34627632108360669882887936617, −4.59095931648610406732832410627, −3.71090613105198694982625741843, −2.52272108200300328109293408851, −1.38615278965778890776976065978, 0,
1.38615278965778890776976065978, 2.52272108200300328109293408851, 3.71090613105198694982625741843, 4.59095931648610406732832410627, 5.34627632108360669882887936617, 6.09693617243412784458763011094, 6.67261145699874479486648339380, 7.66833359605819690013889104167, 8.420071575848007910972502904069
