L(s) = 1 | − 2-s − 3.08·3-s + 4-s − 0.00834·5-s + 3.08·6-s − 0.451·7-s − 8-s + 6.48·9-s + 0.00834·10-s − 3.57·11-s − 3.08·12-s + 3.37·13-s + 0.451·14-s + 0.0256·15-s + 16-s + 1.64·17-s − 6.48·18-s + 1.82·19-s − 0.00834·20-s + 1.39·21-s + 3.57·22-s + 3.38·23-s + 3.08·24-s − 4.99·25-s − 3.37·26-s − 10.7·27-s − 0.451·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.00373·5-s + 1.25·6-s − 0.170·7-s − 0.353·8-s + 2.16·9-s + 0.00263·10-s − 1.07·11-s − 0.889·12-s + 0.937·13-s + 0.120·14-s + 0.00663·15-s + 0.250·16-s + 0.397·17-s − 1.52·18-s + 0.419·19-s − 0.00186·20-s + 0.303·21-s + 0.762·22-s + 0.706·23-s + 0.628·24-s − 0.999·25-s − 0.662·26-s − 2.06·27-s − 0.0852·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5615206845\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5615206845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 5 | \( 1 + 0.00834T + 5T^{2} \) |
| 7 | \( 1 + 0.451T + 7T^{2} \) |
| 11 | \( 1 + 3.57T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 - 0.201T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 + 2.11T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 4.23T + 59T^{2} \) |
| 61 | \( 1 + 2.00T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 - 3.24T + 79T^{2} \) |
| 83 | \( 1 + 2.60T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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
−8.308213556642795604391394662977, −7.63138992022474016947705475199, −6.94953239261396067519141721208, −6.10856795548362819651765026537, −5.70777812953831334605653559952, −4.95274792428955110310719918564, −4.00130678883180541367896132143, −2.81740746489390912609286798375, −1.46805322910689172526787612745, −0.55888990023084356851247044929,
0.55888990023084356851247044929, 1.46805322910689172526787612745, 2.81740746489390912609286798375, 4.00130678883180541367896132143, 4.95274792428955110310719918564, 5.70777812953831334605653559952, 6.10856795548362819651765026537, 6.94953239261396067519141721208, 7.63138992022474016947705475199, 8.308213556642795604391394662977