| L(s) = 1 | − 2.38·2-s + 1.02·3-s + 3.70·4-s − 0.691·5-s − 2.44·6-s + 1.59·7-s − 4.08·8-s − 1.95·9-s + 1.65·10-s − 3.76·11-s + 3.79·12-s − 6.60·13-s − 3.80·14-s − 0.708·15-s + 2.34·16-s − 3.52·17-s + 4.66·18-s + 8.47·19-s − 2.56·20-s + 1.62·21-s + 8.99·22-s − 5.10·23-s − 4.18·24-s − 4.52·25-s + 15.7·26-s − 5.06·27-s + 5.90·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.591·3-s + 1.85·4-s − 0.309·5-s − 0.998·6-s + 0.601·7-s − 1.44·8-s − 0.650·9-s + 0.522·10-s − 1.13·11-s + 1.09·12-s − 1.83·13-s − 1.01·14-s − 0.182·15-s + 0.585·16-s − 0.854·17-s + 1.09·18-s + 1.94·19-s − 0.574·20-s + 0.355·21-s + 1.91·22-s − 1.06·23-s − 0.853·24-s − 0.904·25-s + 3.09·26-s − 0.975·27-s + 1.11·28-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
| \(L(1)\) |
\(\approx\) |
\(0.4596738341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4596738341\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 - 1.02T + 3T^{2} \) |
| 5 | \( 1 + 0.691T + 5T^{2} \) |
| 7 | \( 1 - 1.59T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 6.60T + 13T^{2} \) |
| 17 | \( 1 + 3.52T + 17T^{2} \) |
| 19 | \( 1 - 8.47T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 + 0.0916T + 41T^{2} \) |
| 43 | \( 1 + 0.702T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 3.97T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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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
−8.202141795207500101542127120761, −7.959148048254098783632735897555, −7.50808428261452786644991940297, −6.70452050630646232558798158257, −5.46045807452741756238399519565, −4.86229373788120986367489356269, −3.47298285978335707288731362227, −2.42034670917199876563147372879, −2.06773402979457735084837880485, −0.45360298207032315644978397302,
0.45360298207032315644978397302, 2.06773402979457735084837880485, 2.42034670917199876563147372879, 3.47298285978335707288731362227, 4.86229373788120986367489356269, 5.46045807452741756238399519565, 6.70452050630646232558798158257, 7.50808428261452786644991940297, 7.959148048254098783632735897555, 8.202141795207500101542127120761
