| L(s) = 1 | − 2-s + 4-s − 2.34·5-s + 0.305·7-s − 8-s + 2.34·10-s − 2.70·11-s − 4.89·13-s − 0.305·14-s + 16-s − 7.93·17-s + 2.53·19-s − 2.34·20-s + 2.70·22-s − 5.66·23-s + 0.491·25-s + 4.89·26-s + 0.305·28-s − 1.40·29-s − 5.41·31-s − 32-s + 7.93·34-s − 0.716·35-s + 10.9·37-s − 2.53·38-s + 2.34·40-s − 0.828·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.04·5-s + 0.115·7-s − 0.353·8-s + 0.741·10-s − 0.814·11-s − 1.35·13-s − 0.0817·14-s + 0.250·16-s − 1.92·17-s + 0.581·19-s − 0.524·20-s + 0.575·22-s − 1.18·23-s + 0.0983·25-s + 0.959·26-s + 0.0578·28-s − 0.261·29-s − 0.972·31-s − 0.176·32-s + 1.36·34-s − 0.121·35-s + 1.79·37-s − 0.411·38-s + 0.370·40-s − 0.129·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09027154978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09027154978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 - 0.305T + 7T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 + 4.89T + 13T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 - 2.53T + 19T^{2} \) |
| 23 | \( 1 + 5.66T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 4.88T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 1.25T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 + 8.79T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.17T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
| 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
−7.78152602758510737852036114347, −7.44328609495418249206081666957, −6.66578777594450132506163610140, −5.88074953568601526517816665145, −4.84546244394652875237184242760, −4.41064168250542249416718233268, −3.39561943772852989024705738130, −2.53052741887444374179766682016, −1.80800580102500366421074769971, −0.15479356035186714925286213408,
0.15479356035186714925286213408, 1.80800580102500366421074769971, 2.53052741887444374179766682016, 3.39561943772852989024705738130, 4.41064168250542249416718233268, 4.84546244394652875237184242760, 5.88074953568601526517816665145, 6.66578777594450132506163610140, 7.44328609495418249206081666957, 7.78152602758510737852036114347
