L(s) = 1 | + 2-s + 4-s + 1.29·5-s − 0.779·7-s + 8-s + 1.29·10-s − 2.38·11-s + 0.594·13-s − 0.779·14-s + 16-s − 5.75·17-s + 0.870·19-s + 1.29·20-s − 2.38·22-s + 2.02·23-s − 3.32·25-s + 0.594·26-s − 0.779·28-s + 3.56·29-s − 4.32·31-s + 32-s − 5.75·34-s − 1.00·35-s − 5.50·37-s + 0.870·38-s + 1.29·40-s − 1.77·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.578·5-s − 0.294·7-s + 0.353·8-s + 0.408·10-s − 0.719·11-s + 0.164·13-s − 0.208·14-s + 0.250·16-s − 1.39·17-s + 0.199·19-s + 0.289·20-s − 0.508·22-s + 0.422·23-s − 0.665·25-s + 0.116·26-s − 0.147·28-s + 0.661·29-s − 0.777·31-s + 0.176·32-s − 0.986·34-s − 0.170·35-s − 0.904·37-s + 0.141·38-s + 0.204·40-s − 0.277·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 + 0.779T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 - 0.594T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 - 0.870T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 4.32T + 31T^{2} \) |
| 37 | \( 1 + 5.50T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 + 5.52T + 47T^{2} \) |
| 53 | \( 1 - 0.474T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 - 6.01T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 0.746T + 71T^{2} \) |
| 73 | \( 1 + 9.67T + 73T^{2} \) |
| 79 | \( 1 + 4.04T + 79T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 + 1.20T + 89T^{2} \) |
| 97 | \( 1 - 0.348T + 97T^{2} \) |
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
−7.26412934545670745968301043209, −6.69347601423974300849046979298, −6.04673961658296451317762474631, −5.36247198431348996086502821167, −4.75157208807333951382440437669, −3.94043263988796090070850001608, −3.07897307867997208437588388750, −2.35781902147189453584049878193, −1.55434419022248701766580676549, 0,
1.55434419022248701766580676549, 2.35781902147189453584049878193, 3.07897307867997208437588388750, 3.94043263988796090070850001608, 4.75157208807333951382440437669, 5.36247198431348996086502821167, 6.04673961658296451317762474631, 6.69347601423974300849046979298, 7.26412934545670745968301043209