L(s) = 1 | + 2.68·2-s − 2.98·3-s + 5.19·4-s − 3.56·5-s − 8.00·6-s + 3.18·7-s + 8.55·8-s + 5.91·9-s − 9.55·10-s − 11-s − 15.4·12-s + 5.51·13-s + 8.54·14-s + 10.6·15-s + 12.5·16-s + 1.59·17-s + 15.8·18-s + 0.342·19-s − 18.4·20-s − 9.51·21-s − 2.68·22-s + 8.05·23-s − 25.5·24-s + 7.69·25-s + 14.7·26-s − 8.70·27-s + 16.5·28-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 1.72·3-s + 2.59·4-s − 1.59·5-s − 3.26·6-s + 1.20·7-s + 3.02·8-s + 1.97·9-s − 3.02·10-s − 0.301·11-s − 4.47·12-s + 1.52·13-s + 2.28·14-s + 2.74·15-s + 3.14·16-s + 0.385·17-s + 3.73·18-s + 0.0786·19-s − 4.13·20-s − 2.07·21-s − 0.571·22-s + 1.67·23-s − 5.21·24-s + 1.53·25-s + 2.89·26-s − 1.67·27-s + 3.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.648594960\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.648594960\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 - 0.342T + 19T^{2} \) |
| 23 | \( 1 - 8.05T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + 0.725T + 31T^{2} \) |
| 37 | \( 1 - 0.0605T + 37T^{2} \) |
| 41 | \( 1 + 0.550T + 41T^{2} \) |
| 43 | \( 1 + 7.35T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 67 | \( 1 - 1.98T + 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 + 0.581T + 73T^{2} \) |
| 79 | \( 1 + 4.50T + 79T^{2} \) |
| 83 | \( 1 + 4.28T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 - 8.34T + 97T^{2} \) |
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
−11.19685225612404933958094864968, −10.58767871514895123393496605606, −8.301033297100918563056084866411, −7.35671815145293535845989533879, −6.70892008466413542877175602168, −5.56932158163445238179313049098, −5.05846794040177683672317749926, −4.23796830439172562208685245238, −3.46864203325557860967842362702, −1.31736535991115388956301080897,
1.31736535991115388956301080897, 3.46864203325557860967842362702, 4.23796830439172562208685245238, 5.05846794040177683672317749926, 5.56932158163445238179313049098, 6.70892008466413542877175602168, 7.35671815145293535845989533879, 8.301033297100918563056084866411, 10.58767871514895123393496605606, 11.19685225612404933958094864968