| L(s) = 1 | − 3-s + 1.88·5-s − 1.29·7-s + 9-s − 4.19·11-s − 1.88·15-s − 1.68·17-s − 1.19·19-s + 1.29·21-s − 3.01·23-s − 1.45·25-s − 27-s + 3.15·29-s + 1.47·31-s + 4.19·33-s − 2.43·35-s + 8.15·37-s − 10.0·41-s − 10.7·43-s + 1.88·45-s + 5.31·47-s − 5.33·49-s + 1.68·51-s + 8.29·53-s − 7.89·55-s + 1.19·57-s + 3.03·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.841·5-s − 0.488·7-s + 0.333·9-s − 1.26·11-s − 0.485·15-s − 0.408·17-s − 0.273·19-s + 0.281·21-s − 0.628·23-s − 0.291·25-s − 0.192·27-s + 0.584·29-s + 0.264·31-s + 0.730·33-s − 0.410·35-s + 1.34·37-s − 1.56·41-s − 1.64·43-s + 0.280·45-s + 0.775·47-s − 0.761·49-s + 0.235·51-s + 1.13·53-s − 1.06·55-s + 0.157·57-s + 0.394·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.206965932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.206965932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 1.29T + 7T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 + 1.19T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 - 8.15T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 - 7.84T + 61T^{2} \) |
| 67 | \( 1 + 4.50T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 2.38T + 97T^{2} \) |
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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
−7.941789892974070720180401821529, −6.84578036517093035730216398764, −6.48024249780892130253113002221, −5.69151480526617010222874170844, −5.20479477371435635987320539038, −4.44579578551045484660903494480, −3.46948667523266157644638476313, −2.51929469416784074306513715974, −1.87542465146466671374429171755, −0.53639090342328516579900039334,
0.53639090342328516579900039334, 1.87542465146466671374429171755, 2.51929469416784074306513715974, 3.46948667523266157644638476313, 4.44579578551045484660903494480, 5.20479477371435635987320539038, 5.69151480526617010222874170844, 6.48024249780892130253113002221, 6.84578036517093035730216398764, 7.941789892974070720180401821529
