| L(s) = 1 | − 2.10·2-s + 2.44·4-s − 0.492·5-s + 7-s − 0.948·8-s + 1.03·10-s + 5.30·13-s − 2.10·14-s − 2.89·16-s − 3.03·17-s − 4.66·19-s − 1.20·20-s + 5.63·23-s − 4.75·25-s − 11.1·26-s + 2.44·28-s − 6.92·29-s − 1.26·31-s + 8.01·32-s + 6.40·34-s − 0.492·35-s + 10.8·37-s + 9.84·38-s + 0.466·40-s + 1.44·41-s + 2.88·43-s − 11.8·46-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.22·4-s − 0.220·5-s + 0.377·7-s − 0.335·8-s + 0.328·10-s + 1.47·13-s − 0.563·14-s − 0.724·16-s − 0.736·17-s − 1.07·19-s − 0.269·20-s + 1.17·23-s − 0.951·25-s − 2.19·26-s + 0.462·28-s − 1.28·29-s − 0.227·31-s + 1.41·32-s + 1.09·34-s − 0.0832·35-s + 1.78·37-s + 1.59·38-s + 0.0738·40-s + 0.225·41-s + 0.439·43-s − 1.75·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 5 | \( 1 + 0.492T + 5T^{2} \) |
| 13 | \( 1 - 5.30T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 + 4.66T + 19T^{2} \) |
| 23 | \( 1 - 5.63T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 + 6.63T + 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 + 5.94T + 71T^{2} \) |
| 73 | \( 1 + 3.77T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 3.10T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−7.66004697027783098638778508944, −7.17671638320095793948709939295, −6.26666177764986719414626339153, −5.76273872728203033328489227234, −4.47106105216504100762392272576, −4.03714174429247347846756216014, −2.81889311352473898650731920682, −1.88134794701729154278391089569, −1.13688301700403833279745642239, 0,
1.13688301700403833279745642239, 1.88134794701729154278391089569, 2.81889311352473898650731920682, 4.03714174429247347846756216014, 4.47106105216504100762392272576, 5.76273872728203033328489227234, 6.26666177764986719414626339153, 7.17671638320095793948709939295, 7.66004697027783098638778508944
