| L(s) = 1 | − 2.55·2-s + 2.40·3-s + 4.52·4-s + 3.51·5-s − 6.14·6-s + 3.52·7-s − 6.45·8-s + 2.79·9-s − 8.97·10-s − 5.09·11-s + 10.8·12-s − 0.353·13-s − 9.00·14-s + 8.45·15-s + 7.43·16-s + 1.22·17-s − 7.13·18-s + 0.445·19-s + 15.8·20-s + 8.48·21-s + 13.0·22-s + 9.35·23-s − 15.5·24-s + 7.33·25-s + 0.902·26-s − 0.499·27-s + 15.9·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.38·3-s + 2.26·4-s + 1.57·5-s − 2.51·6-s + 1.33·7-s − 2.28·8-s + 0.930·9-s − 2.83·10-s − 1.53·11-s + 3.14·12-s − 0.0980·13-s − 2.40·14-s + 2.18·15-s + 1.85·16-s + 0.297·17-s − 1.68·18-s + 0.102·19-s + 3.55·20-s + 1.85·21-s + 2.77·22-s + 1.95·23-s − 3.17·24-s + 1.46·25-s + 0.177·26-s − 0.0961·27-s + 3.01·28-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
| \(L(1)\) |
\(\approx\) |
\(2.155927920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155927920\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 0.353T + 13T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 - 0.445T + 19T^{2} \) |
| 23 | \( 1 - 9.35T + 23T^{2} \) |
| 29 | \( 1 + 6.41T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 + 0.0828T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 9.28T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 59 | \( 1 + 7.77T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 - 6.33T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 0.799T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 7.86T + 97T^{2} \) |
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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
−8.528087008990533157779426384185, −7.963713195445481447856198604456, −7.47921456335753742727125960592, −6.66826718135151494555981933656, −5.51876087148877085417732342608, −4.95470688580900844765461650460, −3.16754471653913891324868138958, −2.39347724360903634990767413225, −1.99651556093890350649722812539, −1.09359775502743541431963851287,
1.09359775502743541431963851287, 1.99651556093890350649722812539, 2.39347724360903634990767413225, 3.16754471653913891324868138958, 4.95470688580900844765461650460, 5.51876087148877085417732342608, 6.66826718135151494555981933656, 7.47921456335753742727125960592, 7.963713195445481447856198604456, 8.528087008990533157779426384185
