L(s) = 1 | − 2.33·2-s − 3-s + 3.42·4-s + 0.900·5-s + 2.33·6-s + 7-s − 3.33·8-s + 9-s − 2.09·10-s + 2.90·11-s − 3.42·12-s − 2.33·14-s − 0.900·15-s + 0.900·16-s + 1.66·17-s − 2.33·18-s − 3.66·19-s + 3.08·20-s − 21-s − 6.75·22-s − 4.42·23-s + 3.33·24-s − 4.18·25-s − 27-s + 3.42·28-s + 5.33·29-s + 2.09·30-s + ⋯ |
L(s) = 1 | − 1.64·2-s − 0.577·3-s + 1.71·4-s + 0.402·5-s + 0.951·6-s + 0.377·7-s − 1.17·8-s + 0.333·9-s − 0.663·10-s + 0.874·11-s − 0.989·12-s − 0.622·14-s − 0.232·15-s + 0.225·16-s + 0.405·17-s − 0.549·18-s − 0.839·19-s + 0.690·20-s − 0.218·21-s − 1.44·22-s − 0.923·23-s + 0.679·24-s − 0.837·25-s − 0.192·27-s + 0.648·28-s + 0.989·29-s + 0.383·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7639406552\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7639406552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 5 | \( 1 - 0.900T + 5T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 17 | \( 1 - 1.66T + 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 5.33T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 8.61T + 37T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 + 0.141T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 2.51T + 61T^{2} \) |
| 67 | \( 1 - 0.00982T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + 2.57T + 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 - 1.58T + 83T^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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
−8.483996033354997742403156649173, −8.096688693501292191536325176045, −7.20580301115040640744037313730, −6.42598657102315608020892914493, −6.00960207491049089655841672755, −4.81118236441577375162507507895, −3.93721643436576610062788017529, −2.46566907379832315435448226250, −1.63157801700168671679271050393, −0.71115631222107866255398119243,
0.71115631222107866255398119243, 1.63157801700168671679271050393, 2.46566907379832315435448226250, 3.93721643436576610062788017529, 4.81118236441577375162507507895, 6.00960207491049089655841672755, 6.42598657102315608020892914493, 7.20580301115040640744037313730, 8.096688693501292191536325176045, 8.483996033354997742403156649173