| L(s) = 1 | + 2.77·2-s + 5.72·4-s + 1.30·5-s + 7-s + 10.3·8-s + 3.61·10-s − 1.22·11-s + 1.99·13-s + 2.77·14-s + 17.3·16-s − 3.84·17-s − 7.61·19-s + 7.44·20-s − 3.40·22-s + 5.59·23-s − 3.30·25-s + 5.54·26-s + 5.72·28-s + 1.90·29-s + 4.73·31-s + 27.4·32-s − 10.6·34-s + 1.30·35-s + 7.21·37-s − 21.1·38-s + 13.4·40-s + 6.26·41-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.86·4-s + 0.581·5-s + 0.377·7-s + 3.65·8-s + 1.14·10-s − 0.369·11-s + 0.552·13-s + 0.742·14-s + 4.32·16-s − 0.932·17-s − 1.74·19-s + 1.66·20-s − 0.725·22-s + 1.16·23-s − 0.661·25-s + 1.08·26-s + 1.08·28-s + 0.354·29-s + 0.849·31-s + 4.84·32-s − 1.83·34-s + 0.219·35-s + 1.18·37-s − 3.43·38-s + 2.12·40-s + 0.979·41-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
| \(L(1)\) |
\(\approx\) |
\(9.873971153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.873971153\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.77T + 2T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 - 5.59T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 - 1.73T + 43T^{2} \) |
| 47 | \( 1 - 3.16T + 47T^{2} \) |
| 53 | \( 1 - 0.926T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 8.05T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 8.77T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 - 6.15T + 89T^{2} \) |
| 97 | \( 1 + 9.84T + 97T^{2} \) |
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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
−7.58039332287347079040760000190, −6.69091882644463706841819734187, −6.27941340546235695268320697809, −5.72672247131093974236730358730, −4.89028263665926297159012236491, −4.38780224003074666760069513948, −3.77157354204666757757451101075, −2.58919664070616571647629755429, −2.34373511581381080391420163069, −1.26802055994153194805807363818,
1.26802055994153194805807363818, 2.34373511581381080391420163069, 2.58919664070616571647629755429, 3.77157354204666757757451101075, 4.38780224003074666760069513948, 4.89028263665926297159012236491, 5.72672247131093974236730358730, 6.27941340546235695268320697809, 6.69091882644463706841819734187, 7.58039332287347079040760000190
