| L(s) = 1 | − 1.23·3-s − 1.23·5-s − 7-s − 1.47·9-s − 11-s − 3.23·13-s + 1.52·15-s + 2.47·17-s + 7.23·19-s + 1.23·21-s − 4·23-s − 3.47·25-s + 5.52·27-s + 4.47·29-s − 2·31-s + 1.23·33-s + 1.23·35-s + 6.94·37-s + 4.00·39-s − 2.47·41-s + 10.4·43-s + 1.81·45-s + 2·47-s + 49-s − 3.05·51-s + 8.47·53-s + 1.23·55-s + ⋯ |
| L(s) = 1 | − 0.713·3-s − 0.552·5-s − 0.377·7-s − 0.490·9-s − 0.301·11-s − 0.897·13-s + 0.394·15-s + 0.599·17-s + 1.66·19-s + 0.269·21-s − 0.834·23-s − 0.694·25-s + 1.06·27-s + 0.830·29-s − 0.359·31-s + 0.215·33-s + 0.208·35-s + 1.14·37-s + 0.640·39-s − 0.386·41-s + 1.59·43-s + 0.271·45-s + 0.291·47-s + 0.142·49-s − 0.427·51-s + 1.16·53-s + 0.166·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8204148898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8204148898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.828237200310757091105725418800, −8.993244297100908944381335897944, −7.79742474527576272538489042608, −7.44926801779900773094427088827, −6.21743626570945632252289309022, −5.55700459274069434229157399327, −4.70995222449309253733128721235, −3.54810811370255646987818517922, −2.55622385667059419463073649339, −0.67619814292865392304631893588,
0.67619814292865392304631893588, 2.55622385667059419463073649339, 3.54810811370255646987818517922, 4.70995222449309253733128721235, 5.55700459274069434229157399327, 6.21743626570945632252289309022, 7.44926801779900773094427088827, 7.79742474527576272538489042608, 8.993244297100908944381335897944, 9.828237200310757091105725418800
