L(s) = 1 | + 0.635·2-s + 2.40·3-s − 1.59·4-s − 0.176·5-s + 1.52·6-s + 0.607·7-s − 2.28·8-s + 2.79·9-s − 0.111·10-s + 0.724·11-s − 3.84·12-s − 3.12·13-s + 0.385·14-s − 0.424·15-s + 1.74·16-s − 1.81·17-s + 1.77·18-s + 3.76·19-s + 0.281·20-s + 1.46·21-s + 0.459·22-s + 4.76·23-s − 5.49·24-s − 4.96·25-s − 1.98·26-s − 0.488·27-s − 0.970·28-s + ⋯ |
L(s) = 1 | + 0.449·2-s + 1.39·3-s − 0.798·4-s − 0.0787·5-s + 0.624·6-s + 0.229·7-s − 0.807·8-s + 0.932·9-s − 0.0353·10-s + 0.218·11-s − 1.10·12-s − 0.866·13-s + 0.103·14-s − 0.109·15-s + 0.435·16-s − 0.440·17-s + 0.418·18-s + 0.864·19-s + 0.0628·20-s + 0.319·21-s + 0.0980·22-s + 0.993·23-s − 1.12·24-s − 0.993·25-s − 0.389·26-s − 0.0939·27-s − 0.183·28-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
\(L(1)\) |
\(\approx\) |
\(3.166113003\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.166113003\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.635T + 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 5 | \( 1 + 0.176T + 5T^{2} \) |
| 7 | \( 1 - 0.607T + 7T^{2} \) |
| 11 | \( 1 - 0.724T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 1.81T + 17T^{2} \) |
| 19 | \( 1 - 3.76T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + 8.43T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 3.00T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 7.69T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 0.176T + 89T^{2} \) |
| 97 | \( 1 - 8.88T + 97T^{2} \) |
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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
−8.221427166135394648595867863359, −8.127416957150230157375570345864, −7.11504994753222242353083083768, −6.25248023309007550947657730659, −5.14469145856837819154476398624, −4.59870458424750377646260137709, −3.79947617294834671662097690393, −2.99496538104694308719629905951, −2.37442417410358334100929657818, −0.913941589373625120554315972341,
0.913941589373625120554315972341, 2.37442417410358334100929657818, 2.99496538104694308719629905951, 3.79947617294834671662097690393, 4.59870458424750377646260137709, 5.14469145856837819154476398624, 6.25248023309007550947657730659, 7.11504994753222242353083083768, 8.127416957150230157375570345864, 8.221427166135394648595867863359