| L(s) = 1 | + 0.684·2-s − 1.53·4-s + 0.182·5-s + 7-s − 2.41·8-s + 0.125·10-s − 4.23·11-s + 3.25·13-s + 0.684·14-s + 1.40·16-s + 2.73·17-s + 5.66·19-s − 0.279·20-s − 2.90·22-s − 3.08·23-s − 4.96·25-s + 2.23·26-s − 1.53·28-s + 6.70·29-s − 9.55·31-s + 5.79·32-s + 1.87·34-s + 0.182·35-s − 7.75·37-s + 3.87·38-s − 0.441·40-s − 0.852·41-s + ⋯ |
| L(s) = 1 | + 0.484·2-s − 0.765·4-s + 0.0817·5-s + 0.377·7-s − 0.854·8-s + 0.0395·10-s − 1.27·11-s + 0.903·13-s + 0.183·14-s + 0.351·16-s + 0.662·17-s + 1.29·19-s − 0.0625·20-s − 0.618·22-s − 0.644·23-s − 0.993·25-s + 0.437·26-s − 0.289·28-s + 1.24·29-s − 1.71·31-s + 1.02·32-s + 0.320·34-s + 0.0308·35-s − 1.27·37-s + 0.629·38-s − 0.0698·40-s − 0.133·41-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 | \( 1 - 0.684T + 2T^{2} \) |
| 5 | \( 1 - 0.182T + 5T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 3.25T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 - 5.66T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 + 0.852T + 41T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 + 5.27T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 - 7.70T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 - 8.52T + 67T^{2} \) |
| 71 | \( 1 - 7.08T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 9.91T + 89T^{2} \) |
| 97 | \( 1 + 7.39T + 97T^{2} \) |
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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
−7.71521325710674324120442376007, −6.70471342985878241439591383450, −5.80996373839024945754842643266, −5.26401244638477876124517553205, −4.92896395930628957391134422141, −3.66865874819562462465959198764, −3.49175239926773392864875737278, −2.33516429388552157358540577967, −1.22756179373740320408204349124, 0,
1.22756179373740320408204349124, 2.33516429388552157358540577967, 3.49175239926773392864875737278, 3.66865874819562462465959198764, 4.92896395930628957391134422141, 5.26401244638477876124517553205, 5.80996373839024945754842643266, 6.70471342985878241439591383450, 7.71521325710674324120442376007
