| L(s) = 1 | + 2.27·2-s + 3-s + 3.17·4-s − 2.39·5-s + 2.27·6-s + 2.66·8-s + 9-s − 5.44·10-s − 4.56·11-s + 3.17·12-s − 5.05·13-s − 2.39·15-s − 0.274·16-s + 1.44·17-s + 2.27·18-s − 1.98·19-s − 7.60·20-s − 10.3·22-s − 23-s + 2.66·24-s + 0.737·25-s − 11.4·26-s + 27-s + 3.68·29-s − 5.44·30-s + 3.44·31-s − 5.96·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.58·4-s − 1.07·5-s + 0.928·6-s + 0.943·8-s + 0.333·9-s − 1.72·10-s − 1.37·11-s + 0.916·12-s − 1.40·13-s − 0.618·15-s − 0.0686·16-s + 0.351·17-s + 0.536·18-s − 0.454·19-s − 1.69·20-s − 2.21·22-s − 0.208·23-s + 0.544·24-s + 0.147·25-s − 2.25·26-s + 0.192·27-s + 0.685·29-s − 0.994·30-s + 0.619·31-s − 1.05·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 29 | \( 1 - 3.68T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 3.99T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 + 1.25T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 8.37T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 + 0.946T + 71T^{2} \) |
| 73 | \( 1 + 3.86T + 73T^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 0.262T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−7.939152662878256910562222358360, −7.47094403941771536542581421003, −6.73381048247869307133117703557, −5.73141874531871809251266855853, −4.82007220982810122412169922035, −4.53410876018437973770175929554, −3.49230347163227219109367895861, −2.89795394102340932445339098090, −2.12731864060647501082596650489, 0,
2.12731864060647501082596650489, 2.89795394102340932445339098090, 3.49230347163227219109367895861, 4.53410876018437973770175929554, 4.82007220982810122412169922035, 5.73141874531871809251266855853, 6.73381048247869307133117703557, 7.47094403941771536542581421003, 7.939152662878256910562222358360
