L(s) = 1 | + 0.154·3-s − 1.21·5-s − 0.213·7-s − 2.97·9-s + 6.43·11-s + 5.00·13-s − 0.187·15-s − 17-s − 5.48·19-s − 0.0329·21-s + 0.395·23-s − 3.53·25-s − 0.925·27-s + 0.149·29-s + 6.05·31-s + 0.996·33-s + 0.257·35-s + 9.39·37-s + 0.775·39-s + 3.27·41-s − 10.2·43-s + 3.60·45-s + 5.33·47-s − 6.95·49-s − 0.154·51-s − 1.89·53-s − 7.78·55-s + ⋯ |
L(s) = 1 | + 0.0894·3-s − 0.541·5-s − 0.0805·7-s − 0.992·9-s + 1.93·11-s + 1.38·13-s − 0.0484·15-s − 0.242·17-s − 1.25·19-s − 0.00719·21-s + 0.0824·23-s − 0.706·25-s − 0.178·27-s + 0.0277·29-s + 1.08·31-s + 0.173·33-s + 0.0435·35-s + 1.54·37-s + 0.124·39-s + 0.510·41-s − 1.56·43-s + 0.537·45-s + 0.778·47-s − 0.993·49-s − 0.0216·51-s − 0.259·53-s − 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.788467264\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788467264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 0.154T + 3T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 + 0.213T + 7T^{2} \) |
| 11 | \( 1 - 6.43T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 - 0.395T + 23T^{2} \) |
| 29 | \( 1 - 0.149T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 5.33T + 47T^{2} \) |
| 53 | \( 1 + 1.89T + 53T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 4.18T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.352426415304362382067842680020, −8.051039131712225420318619838150, −6.60982046546254828566127640600, −6.47645292122555797911529056299, −5.65784961402462751337343449125, −4.35646338486339133873665911581, −3.92775114270324671616579918304, −3.12018131859035507646961972729, −1.91935929488330592453683447220, −0.77839752617304777074181101413,
0.77839752617304777074181101413, 1.91935929488330592453683447220, 3.12018131859035507646961972729, 3.92775114270324671616579918304, 4.35646338486339133873665911581, 5.65784961402462751337343449125, 6.47645292122555797911529056299, 6.60982046546254828566127640600, 8.051039131712225420318619838150, 8.352426415304362382067842680020