| L(s) = 1 | + 1.18·2-s − 0.607·4-s − 4.43·5-s − 7-s − 3.07·8-s − 5.23·10-s + 5.41·11-s − 2.38·13-s − 1.18·14-s − 2.41·16-s − 2.36·17-s − 2.75·19-s + 2.69·20-s + 6.39·22-s + 5.73·23-s + 14.6·25-s − 2.81·26-s + 0.607·28-s + 3.90·29-s + 2.83·31-s + 3.30·32-s − 2.78·34-s + 4.43·35-s − 3.58·37-s − 3.25·38-s + 13.6·40-s + 0.707·41-s + ⋯ |
| L(s) = 1 | + 0.834·2-s − 0.303·4-s − 1.98·5-s − 0.377·7-s − 1.08·8-s − 1.65·10-s + 1.63·11-s − 0.661·13-s − 0.315·14-s − 0.604·16-s − 0.572·17-s − 0.632·19-s + 0.602·20-s + 1.36·22-s + 1.19·23-s + 2.93·25-s − 0.551·26-s + 0.114·28-s + 0.725·29-s + 0.509·31-s + 0.583·32-s − 0.477·34-s + 0.750·35-s − 0.589·37-s − 0.527·38-s + 2.15·40-s + 0.110·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 5 | \( 1 + 4.43T + 5T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 - 0.707T + 41T^{2} \) |
| 43 | \( 1 - 9.07T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 - 5.85T + 83T^{2} \) |
| 89 | \( 1 + 5.77T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−7.32207849325173347988493187989, −6.74230761037801094874118285583, −6.25067062416592753409078402220, −4.99585450904242925090480185116, −4.51755211721475024918545482587, −3.99877247238533107699386530650, −3.40791057096362434021243319148, −2.70845072605398084819508661890, −0.992384413325663856364827561683, 0,
0.992384413325663856364827561683, 2.70845072605398084819508661890, 3.40791057096362434021243319148, 3.99877247238533107699386530650, 4.51755211721475024918545482587, 4.99585450904242925090480185116, 6.25067062416592753409078402220, 6.74230761037801094874118285583, 7.32207849325173347988493187989
