| L(s) = 1 | − 2.61·2-s + 2.84·3-s + 4.83·4-s + 0.542·5-s − 7.44·6-s + 4.00·7-s − 7.41·8-s + 5.11·9-s − 1.41·10-s + 4.26·11-s + 13.7·12-s + 5.82·13-s − 10.4·14-s + 1.54·15-s + 9.71·16-s − 4.23·17-s − 13.3·18-s + 6.94·19-s + 2.62·20-s + 11.4·21-s − 11.1·22-s − 5.99·23-s − 21.1·24-s − 4.70·25-s − 15.2·26-s + 6.01·27-s + 19.3·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 1.64·3-s + 2.41·4-s + 0.242·5-s − 3.04·6-s + 1.51·7-s − 2.62·8-s + 1.70·9-s − 0.448·10-s + 1.28·11-s + 3.97·12-s + 1.61·13-s − 2.79·14-s + 0.398·15-s + 2.42·16-s − 1.02·17-s − 3.15·18-s + 1.59·19-s + 0.586·20-s + 2.48·21-s − 2.37·22-s − 1.25·23-s − 4.31·24-s − 0.941·25-s − 2.98·26-s + 1.15·27-s + 3.66·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.357857985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357857985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4001 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.84T + 3T^{2} \) |
| 5 | \( 1 - 0.542T + 5T^{2} \) |
| 7 | \( 1 - 4.00T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 - 5.82T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 + 5.99T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 - 9.01T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 - 5.31T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 7.82T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 + 4.77T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.486660183358852980999576906244, −7.978398572752588591225072055759, −7.58368016532377087964816709308, −6.63480236728174877918278112908, −5.88751309773272995410400757885, −4.33183739836957037295187293740, −3.56207373707181882319408347555, −2.48149903705243565732996765624, −1.57954188456051847488092121893, −1.33394055734303340366189478329,
1.33394055734303340366189478329, 1.57954188456051847488092121893, 2.48149903705243565732996765624, 3.56207373707181882319408347555, 4.33183739836957037295187293740, 5.88751309773272995410400757885, 6.63480236728174877918278112908, 7.58368016532377087964816709308, 7.978398572752588591225072055759, 8.486660183358852980999576906244
