| L(s) = 1 | + 2.70·2-s + 5.31·4-s + 2.83·5-s + 2.54·7-s + 8.94·8-s + 7.66·10-s + 11-s − 3.40·13-s + 6.87·14-s + 13.5·16-s − 3.34·17-s + 0.886·19-s + 15.0·20-s + 2.70·22-s + 2.73·23-s + 3.03·25-s − 9.21·26-s + 13.5·28-s − 2.08·29-s − 10.1·31-s + 18.8·32-s − 9.05·34-s + 7.21·35-s + 9.14·37-s + 2.39·38-s + 25.3·40-s − 8.50·41-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.65·4-s + 1.26·5-s + 0.961·7-s + 3.16·8-s + 2.42·10-s + 0.301·11-s − 0.945·13-s + 1.83·14-s + 3.39·16-s − 0.812·17-s + 0.203·19-s + 3.36·20-s + 0.576·22-s + 0.569·23-s + 0.607·25-s − 1.80·26-s + 2.55·28-s − 0.387·29-s − 1.81·31-s + 3.32·32-s − 1.55·34-s + 1.21·35-s + 1.50·37-s + 0.388·38-s + 4.01·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.20611254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.20611254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 2.54T + 7T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 - 0.886T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 9.14T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 + 3.26T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 - 2.19T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 8.61T + 73T^{2} \) |
| 79 | \( 1 + 0.270T + 79T^{2} \) |
| 83 | \( 1 - 8.98T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.63507183407861570644194820286, −7.09844167911643534723515815026, −6.41482724386837016648310578713, −5.63854605661949656743367620747, −5.21988751426875406312742385591, −4.60346993290910606241990219610, −3.84166507092180442565262359357, −2.78873975789276119125096863099, −2.10850469003536770438633925949, −1.53398406861067275016455142801,
1.53398406861067275016455142801, 2.10850469003536770438633925949, 2.78873975789276119125096863099, 3.84166507092180442565262359357, 4.60346993290910606241990219610, 5.21988751426875406312742385591, 5.63854605661949656743367620747, 6.41482724386837016648310578713, 7.09844167911643534723515815026, 7.63507183407861570644194820286
