| L(s) = 1 | + 2.19·3-s + 5-s + 0.544·7-s + 1.81·9-s + 1.63·13-s + 2.19·15-s + 0.169·17-s + 4.72·19-s + 1.19·21-s + 2.26·23-s + 25-s − 2.60·27-s + 9.43·29-s + 3.01·31-s + 0.544·35-s − 0.0537·37-s + 3.59·39-s − 2.54·41-s + 7.46·43-s + 1.81·45-s − 9.07·47-s − 6.70·49-s + 0.370·51-s − 2.03·53-s + 10.3·57-s + 12.6·59-s + 2.80·61-s + ⋯ |
| L(s) = 1 | + 1.26·3-s + 0.447·5-s + 0.205·7-s + 0.603·9-s + 0.454·13-s + 0.566·15-s + 0.0409·17-s + 1.08·19-s + 0.260·21-s + 0.471·23-s + 0.200·25-s − 0.501·27-s + 1.75·29-s + 0.540·31-s + 0.0919·35-s − 0.00882·37-s + 0.575·39-s − 0.397·41-s + 1.13·43-s + 0.270·45-s − 1.32·47-s − 0.957·49-s + 0.0519·51-s − 0.279·53-s + 1.37·57-s + 1.65·59-s + 0.359·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.288300297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288300297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 7 | \( 1 - 0.544T + 7T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 - 0.169T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 - 9.43T + 29T^{2} \) |
| 31 | \( 1 - 3.01T + 31T^{2} \) |
| 37 | \( 1 + 0.0537T + 37T^{2} \) |
| 41 | \( 1 + 2.54T + 41T^{2} \) |
| 43 | \( 1 - 7.46T + 43T^{2} \) |
| 47 | \( 1 + 9.07T + 47T^{2} \) |
| 53 | \( 1 + 2.03T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 2.80T + 61T^{2} \) |
| 67 | \( 1 - 1.26T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 7.97T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 2.73T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−7.85573360902701250730206263459, −7.08669443138321823485192076528, −6.43537990784872008196409833332, −5.59813905902168839793310999836, −4.87569400824977694059254090420, −4.07420728462973788136728504691, −3.14725577371695867566708786319, −2.80292111403127588744266631508, −1.82714445257349625185965010198, −0.975872508083668805537476205201,
0.975872508083668805537476205201, 1.82714445257349625185965010198, 2.80292111403127588744266631508, 3.14725577371695867566708786319, 4.07420728462973788136728504691, 4.87569400824977694059254090420, 5.59813905902168839793310999836, 6.43537990784872008196409833332, 7.08669443138321823485192076528, 7.85573360902701250730206263459
