L(s) = 1 | + 1.86·2-s + 1.48·4-s − 2.52·5-s + 7-s − 0.959·8-s − 4.71·10-s − 2.14·11-s + 3.10·13-s + 1.86·14-s − 4.76·16-s − 5.91·17-s + 5.60·19-s − 3.75·20-s − 4.00·22-s + 7.11·23-s + 1.37·25-s + 5.79·26-s + 1.48·28-s + 7.43·29-s + 5.17·31-s − 6.97·32-s − 11.0·34-s − 2.52·35-s − 4.49·37-s + 10.4·38-s + 2.42·40-s + 6.07·41-s + ⋯ |
L(s) = 1 | + 1.32·2-s + 0.743·4-s − 1.12·5-s + 0.377·7-s − 0.339·8-s − 1.49·10-s − 0.647·11-s + 0.861·13-s + 0.499·14-s − 1.19·16-s − 1.43·17-s + 1.28·19-s − 0.839·20-s − 0.854·22-s + 1.48·23-s + 0.275·25-s + 1.13·26-s + 0.280·28-s + 1.38·29-s + 0.929·31-s − 1.23·32-s − 1.89·34-s − 0.426·35-s − 0.739·37-s + 1.69·38-s + 0.382·40-s + 0.948·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 + 2.14T + 11T^{2} \) |
| 13 | \( 1 - 3.10T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 - 5.17T + 31T^{2} \) |
| 37 | \( 1 + 4.49T + 37T^{2} \) |
| 41 | \( 1 - 6.07T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 4.57T + 61T^{2} \) |
| 67 | \( 1 - 3.13T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 + 1.03T + 97T^{2} \) |
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
−7.34945585393246792738100450621, −6.66829601742486727863228844444, −6.02284045400458484654754232230, −5.05154794879461100635552710377, −4.68237671987111073706130039084, −4.07855636152497597650390001804, −3.11922559395957217898599735973, −2.83327723156707684121577498951, −1.34902644505515969270518934333, 0,
1.34902644505515969270518934333, 2.83327723156707684121577498951, 3.11922559395957217898599735973, 4.07855636152497597650390001804, 4.68237671987111073706130039084, 5.05154794879461100635552710377, 6.02284045400458484654754232230, 6.66829601742486727863228844444, 7.34945585393246792738100450621