| L(s) = 1 | + 1.93·2-s + 1.73·4-s − 3.73·5-s + 0.466·7-s − 0.504·8-s − 7.22·10-s + 5.31·11-s − 1.11·13-s + 0.902·14-s − 4.45·16-s + 3.25·17-s + 0.0992·19-s − 6.49·20-s + 10.2·22-s − 3.59·23-s + 8.97·25-s − 2.15·26-s + 0.811·28-s − 4.76·29-s − 3.11·31-s − 7.60·32-s + 6.30·34-s − 1.74·35-s − 9.77·37-s + 0.191·38-s + 1.88·40-s − 41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.869·4-s − 1.67·5-s + 0.176·7-s − 0.178·8-s − 2.28·10-s + 1.60·11-s − 0.309·13-s + 0.241·14-s − 1.11·16-s + 0.790·17-s + 0.0227·19-s − 1.45·20-s + 2.19·22-s − 0.749·23-s + 1.79·25-s − 0.423·26-s + 0.153·28-s − 0.885·29-s − 0.559·31-s − 1.34·32-s + 1.08·34-s − 0.294·35-s − 1.60·37-s + 0.0311·38-s + 0.298·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 - 0.466T + 7T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 - 0.0992T + 19T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 + 9.77T + 37T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + 3.33T + 47T^{2} \) |
| 53 | \( 1 - 6.33T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 + 5.81T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 8.00T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.52T + 83T^{2} \) |
| 89 | \( 1 - 4.21T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−8.163379496736209571194830914684, −7.26218875886241673085823182309, −6.77346000730922605860972906371, −5.82613251597930186418830675748, −4.98085779306724823835925358531, −4.21079499033393550125385176346, −3.69208705997410828218837778504, −3.18334990253124703497801695138, −1.64642161582078662284835503074, 0,
1.64642161582078662284835503074, 3.18334990253124703497801695138, 3.69208705997410828218837778504, 4.21079499033393550125385176346, 4.98085779306724823835925358531, 5.82613251597930186418830675748, 6.77346000730922605860972906371, 7.26218875886241673085823182309, 8.163379496736209571194830914684
