| L(s) = 1 | + 2.35·2-s + 3.52·4-s + 2.01·5-s − 7-s + 3.59·8-s + 4.73·10-s − 0.0911·11-s − 2.36·13-s − 2.35·14-s + 1.39·16-s + 4.89·17-s − 0.257·19-s + 7.11·20-s − 0.214·22-s − 1.91·23-s − 0.938·25-s − 5.56·26-s − 3.52·28-s + 5.21·29-s + 2.91·31-s − 3.90·32-s + 11.5·34-s − 2.01·35-s + 7.61·37-s − 0.605·38-s + 7.24·40-s + 4.73·41-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.76·4-s + 0.901·5-s − 0.377·7-s + 1.27·8-s + 1.49·10-s − 0.0274·11-s − 0.656·13-s − 0.628·14-s + 0.349·16-s + 1.18·17-s − 0.0590·19-s + 1.59·20-s − 0.0456·22-s − 0.398·23-s − 0.187·25-s − 1.09·26-s − 0.666·28-s + 0.967·29-s + 0.522·31-s − 0.690·32-s + 1.97·34-s − 0.340·35-s + 1.25·37-s − 0.0981·38-s + 1.14·40-s + 0.740·41-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
| \(L(1)\) |
\(\approx\) |
\(6.774070422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.774070422\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 11 | \( 1 + 0.0911T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 0.257T + 19T^{2} \) |
| 23 | \( 1 + 1.91T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 - 9.38T + 43T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 - 5.28T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−7.55316730168415809476759114385, −6.82834302375581896934009998619, −6.17143726223725219321354031318, −5.64424141545732595885272359670, −5.16656377154826952934461691513, −4.25900373717889471422668605731, −3.69895176972822176957606763763, −2.56857284963762477871022669353, −2.43630991805763521136212308948, −1.02436193112801808187750214277,
1.02436193112801808187750214277, 2.43630991805763521136212308948, 2.56857284963762477871022669353, 3.69895176972822176957606763763, 4.25900373717889471422668605731, 5.16656377154826952934461691513, 5.64424141545732595885272359670, 6.17143726223725219321354031318, 6.82834302375581896934009998619, 7.55316730168415809476759114385
