L(s) = 1 | + 0.799·2-s − 0.343·3-s − 1.36·4-s + 3.87·5-s − 0.274·6-s + 3.67·7-s − 2.68·8-s − 2.88·9-s + 3.09·10-s + 3.19·11-s + 0.467·12-s − 5.30·13-s + 2.93·14-s − 1.33·15-s + 0.575·16-s − 1.86·17-s − 2.30·18-s − 2.90·19-s − 5.27·20-s − 1.26·21-s + 2.55·22-s − 3.28·23-s + 0.923·24-s + 10.0·25-s − 4.23·26-s + 2.02·27-s − 4.99·28-s + ⋯ |
L(s) = 1 | + 0.565·2-s − 0.198·3-s − 0.680·4-s + 1.73·5-s − 0.112·6-s + 1.38·7-s − 0.949·8-s − 0.960·9-s + 0.979·10-s + 0.962·11-s + 0.135·12-s − 1.47·13-s + 0.783·14-s − 0.343·15-s + 0.143·16-s − 0.453·17-s − 0.542·18-s − 0.666·19-s − 1.17·20-s − 0.275·21-s + 0.543·22-s − 0.684·23-s + 0.188·24-s + 2.00·25-s − 0.831·26-s + 0.389·27-s − 0.944·28-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
\(L(1)\) |
\(\approx\) |
\(2.891358865\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.891358865\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.799T + 2T^{2} \) |
| 3 | \( 1 + 0.343T + 3T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 6.65T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 - 5.64T + 73T^{2} \) |
| 79 | \( 1 + 7.91T + 79T^{2} \) |
| 83 | \( 1 - 0.854T + 83T^{2} \) |
| 89 | \( 1 + 8.87T + 89T^{2} \) |
| 97 | \( 1 - 6.76T + 97T^{2} \) |
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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
−8.780073311396209564318381208603, −7.80505573000856290891969715068, −6.65360441435251370153750653804, −5.97234235687817244356235595963, −5.47690737362185499590690517674, −4.70521529424060465381552617697, −4.27445532800415361448489526314, −2.68335254251592676766043791362, −2.21002558953799022561149195323, −0.933658214600414393494774710756,
0.933658214600414393494774710756, 2.21002558953799022561149195323, 2.68335254251592676766043791362, 4.27445532800415361448489526314, 4.70521529424060465381552617697, 5.47690737362185499590690517674, 5.97234235687817244356235595963, 6.65360441435251370153750653804, 7.80505573000856290891969715068, 8.780073311396209564318381208603