| L(s) = 1 | − 2.72·2-s + 5.44·4-s + 0.297·5-s − 7-s − 9.39·8-s − 0.811·10-s + 1.00·11-s + 6.53·13-s + 2.72·14-s + 14.7·16-s + 0.490·17-s + 1.20·19-s + 1.61·20-s − 2.74·22-s − 5.71·23-s − 4.91·25-s − 17.8·26-s − 5.44·28-s + 2.54·29-s + 1.43·31-s − 21.4·32-s − 1.33·34-s − 0.297·35-s − 4.61·37-s − 3.27·38-s − 2.79·40-s − 7.46·41-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.72·4-s + 0.133·5-s − 0.377·7-s − 3.32·8-s − 0.256·10-s + 0.303·11-s + 1.81·13-s + 0.729·14-s + 3.68·16-s + 0.118·17-s + 0.275·19-s + 0.362·20-s − 0.585·22-s − 1.19·23-s − 0.982·25-s − 3.49·26-s − 1.02·28-s + 0.472·29-s + 0.256·31-s − 3.78·32-s − 0.229·34-s − 0.0502·35-s − 0.759·37-s − 0.531·38-s − 0.441·40-s − 1.16·41-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
| \(L(1)\) |
\(\approx\) |
\(0.8146569699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8146569699\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 - 0.297T + 5T^{2} \) |
| 11 | \( 1 - 1.00T + 11T^{2} \) |
| 13 | \( 1 - 6.53T + 13T^{2} \) |
| 17 | \( 1 - 0.490T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 + 5.71T + 23T^{2} \) |
| 29 | \( 1 - 2.54T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 + 7.46T + 41T^{2} \) |
| 43 | \( 1 - 2.51T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 2.52T + 59T^{2} \) |
| 61 | \( 1 - 6.71T + 61T^{2} \) |
| 67 | \( 1 - 4.35T + 67T^{2} \) |
| 71 | \( 1 - 9.33T + 71T^{2} \) |
| 73 | \( 1 + 4.25T + 73T^{2} \) |
| 79 | \( 1 - 5.38T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−8.038917322590569717988571297090, −7.37508540051463525424672949518, −6.50811004108831229468856380345, −6.21161675890624859317578207618, −5.47388034197722145017406709260, −3.90554680039357457040967727349, −3.32332354821146817244752387958, −2.23861232012687887325677957866, −1.50999454002282505127216503977, −0.62648987310689977054810692959,
0.62648987310689977054810692959, 1.50999454002282505127216503977, 2.23861232012687887325677957866, 3.32332354821146817244752387958, 3.90554680039357457040967727349, 5.47388034197722145017406709260, 6.21161675890624859317578207618, 6.50811004108831229468856380345, 7.37508540051463525424672949518, 8.038917322590569717988571297090
