| L(s) = 1 | + 0.512·2-s − 1.73·4-s − 1.76·5-s − 7-s − 1.91·8-s − 0.906·10-s + 4.52·11-s + 4.50·13-s − 0.512·14-s + 2.49·16-s − 5.81·17-s + 6.77·19-s + 3.07·20-s + 2.31·22-s + 0.708·23-s − 1.86·25-s + 2.30·26-s + 1.73·28-s − 8.00·29-s + 8.44·31-s + 5.10·32-s − 2.97·34-s + 1.76·35-s + 1.40·37-s + 3.47·38-s + 3.38·40-s + 3.06·41-s + ⋯ |
| L(s) = 1 | + 0.362·2-s − 0.868·4-s − 0.791·5-s − 0.377·7-s − 0.676·8-s − 0.286·10-s + 1.36·11-s + 1.24·13-s − 0.136·14-s + 0.623·16-s − 1.41·17-s + 1.55·19-s + 0.687·20-s + 0.493·22-s + 0.147·23-s − 0.373·25-s + 0.452·26-s + 0.328·28-s − 1.48·29-s + 1.51·31-s + 0.902·32-s − 0.511·34-s + 0.299·35-s + 0.231·37-s + 0.563·38-s + 0.535·40-s + 0.478·41-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
| \(L(1)\) |
\(\approx\) |
\(1.475898981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475898981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.512T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 + 5.81T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 23 | \( 1 - 0.708T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 - 8.44T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 - 3.06T + 41T^{2} \) |
| 43 | \( 1 + 3.92T + 43T^{2} \) |
| 47 | \( 1 + 9.77T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 9.67T + 67T^{2} \) |
| 71 | \( 1 + 7.71T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−7.87496665352593478846465833995, −7.13717779333133311880819854576, −6.24671237911139274596158073184, −5.91251928870157546258999684947, −4.76782628733245763685631981774, −4.26442875526173566419134236999, −3.58270460548253855022657581567, −3.14328311999750518231514827601, −1.59732379011600492155970172872, −0.60189053639297193078665386546,
0.60189053639297193078665386546, 1.59732379011600492155970172872, 3.14328311999750518231514827601, 3.58270460548253855022657581567, 4.26442875526173566419134236999, 4.76782628733245763685631981774, 5.91251928870157546258999684947, 6.24671237911139274596158073184, 7.13717779333133311880819854576, 7.87496665352593478846465833995
