| L(s) = 1 | − 1.63·2-s + 0.660·4-s − 1.75·5-s + 5.06·7-s + 2.18·8-s + 2.86·10-s − 1.08·11-s + 3.01·13-s − 8.25·14-s − 4.88·16-s − 2.47·17-s − 7.12·19-s − 1.15·20-s + 1.76·22-s − 5.33·23-s − 1.92·25-s − 4.91·26-s + 3.34·28-s + 6.80·29-s − 8.37·31-s + 3.60·32-s + 4.04·34-s − 8.87·35-s − 2.09·37-s + 11.6·38-s − 3.83·40-s + 10.6·41-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.330·4-s − 0.784·5-s + 1.91·7-s + 0.772·8-s + 0.904·10-s − 0.325·11-s + 0.835·13-s − 2.20·14-s − 1.22·16-s − 0.600·17-s − 1.63·19-s − 0.259·20-s + 0.376·22-s − 1.11·23-s − 0.384·25-s − 0.963·26-s + 0.632·28-s + 1.26·29-s − 1.50·31-s + 0.636·32-s + 0.692·34-s − 1.50·35-s − 0.344·37-s + 1.88·38-s − 0.605·40-s + 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 + 4.90T + 47T^{2} \) |
| 53 | \( 1 - 4.04T + 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 0.733T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 + 9.62T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.467528780946792976166545541153, −8.130083810344137849068329912224, −7.61396705010621938671782222741, −6.61785045482534643916443982885, −5.44825783825596621601101211787, −4.38648905040956281403626074953, −4.09300391211816780445387853153, −2.21890982838963231072093991462, −1.43582620897821806579359510024, 0,
1.43582620897821806579359510024, 2.21890982838963231072093991462, 4.09300391211816780445387853153, 4.38648905040956281403626074953, 5.44825783825596621601101211787, 6.61785045482534643916443982885, 7.61396705010621938671782222741, 8.130083810344137849068329912224, 8.467528780946792976166545541153
