| L(s) = 1 | + 2.14·2-s + 2.62·4-s + 3.73·5-s + 1.33·8-s + 8.01·10-s − 0.528·11-s − 13-s − 2.37·16-s + 3.80·17-s + 3.88·19-s + 9.77·20-s − 1.13·22-s + 2.43·23-s + 8.91·25-s − 2.14·26-s + 3.56·29-s − 4.99·31-s − 7.77·32-s + 8.17·34-s + 9.45·37-s + 8.34·38-s + 4.97·40-s − 5.21·41-s + 10.3·43-s − 1.38·44-s + 5.23·46-s − 11.5·47-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.31·4-s + 1.66·5-s + 0.471·8-s + 2.53·10-s − 0.159·11-s − 0.277·13-s − 0.593·16-s + 0.922·17-s + 0.890·19-s + 2.18·20-s − 0.242·22-s + 0.507·23-s + 1.78·25-s − 0.421·26-s + 0.662·29-s − 0.896·31-s − 1.37·32-s + 1.40·34-s + 1.55·37-s + 1.35·38-s + 0.787·40-s − 0.814·41-s + 1.57·43-s − 0.208·44-s + 0.772·46-s − 1.68·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.806779368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.806779368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 + 0.528T + 11T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 - 3.88T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 5.21T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 + 7.62T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 - 6.15T + 67T^{2} \) |
| 71 | \( 1 - 6.88T + 71T^{2} \) |
| 73 | \( 1 - 6.45T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 + 2.16T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.925792919861929979226098080280, −7.05525960288391881085652367815, −6.37042900790723861580314135628, −5.78436188823863674464893622438, −5.23394964623955284635218165905, −4.76068846807184883131917197192, −3.63322134009075878646436039976, −2.88315902603493622767308846013, −2.24002803327251798900463769619, −1.19729032305136278447643928668,
1.19729032305136278447643928668, 2.24002803327251798900463769619, 2.88315902603493622767308846013, 3.63322134009075878646436039976, 4.76068846807184883131917197192, 5.23394964623955284635218165905, 5.78436188823863674464893622438, 6.37042900790723861580314135628, 7.05525960288391881085652367815, 7.925792919861929979226098080280
