| L(s) = 1 | − 2.66·2-s + 0.260·3-s + 5.11·4-s − 1.12·5-s − 0.695·6-s − 1.67·7-s − 8.29·8-s − 2.93·9-s + 3.01·10-s + 4.71·11-s + 1.33·12-s − 4.92·13-s + 4.45·14-s − 0.294·15-s + 11.9·16-s − 3.62·17-s + 7.81·18-s − 4.61·19-s − 5.77·20-s − 0.436·21-s − 12.5·22-s − 5.48·23-s − 2.16·24-s − 3.72·25-s + 13.1·26-s − 1.54·27-s − 8.54·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.150·3-s + 2.55·4-s − 0.505·5-s − 0.284·6-s − 0.631·7-s − 2.93·8-s − 0.977·9-s + 0.952·10-s + 1.42·11-s + 0.384·12-s − 1.36·13-s + 1.19·14-s − 0.0760·15-s + 2.97·16-s − 0.878·17-s + 1.84·18-s − 1.05·19-s − 1.29·20-s − 0.0951·21-s − 2.67·22-s − 1.14·23-s − 0.441·24-s − 0.744·25-s + 2.57·26-s − 0.297·27-s − 1.61·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1999016939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1999016939\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 0.260T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 0.539T + 43T^{2} \) |
| 47 | \( 1 + 6.16T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 4.98T + 59T^{2} \) |
| 61 | \( 1 - 9.33T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 4.18T + 71T^{2} \) |
| 73 | \( 1 - 5.46T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 8.72T + 83T^{2} \) |
| 89 | \( 1 + 5.76T + 89T^{2} \) |
| 97 | \( 1 + 3.07T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.411170295814620286376469442647, −8.086644295221762940632406449189, −7.12577894682731818512084706481, −6.48126153975119537587426051900, −6.10146496931437306985216018841, −4.58295784213804151141113084065, −3.50875036982810636676698076405, −2.56505935219990341350092710298, −1.82522440949664043798225691645, −0.31732673065205618647713652731,
0.31732673065205618647713652731, 1.82522440949664043798225691645, 2.56505935219990341350092710298, 3.50875036982810636676698076405, 4.58295784213804151141113084065, 6.10146496931437306985216018841, 6.48126153975119537587426051900, 7.12577894682731818512084706481, 8.086644295221762940632406449189, 8.411170295814620286376469442647
