| L(s) = 1 | − 2-s + 4-s − 1.65·5-s − 2.41·7-s − 8-s + 1.65·10-s − 0.467·11-s − 6.22·13-s + 2.41·14-s + 16-s − 5.90·19-s − 1.65·20-s + 0.467·22-s − 1.92·23-s − 2.26·25-s + 6.22·26-s − 2.41·28-s − 4.90·29-s + 0.837·31-s − 32-s + 3.98·35-s + 1.16·37-s + 5.90·38-s + 1.65·40-s − 4.04·41-s − 2.65·43-s − 0.467·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.739·5-s − 0.911·7-s − 0.353·8-s + 0.522·10-s − 0.141·11-s − 1.72·13-s + 0.644·14-s + 0.250·16-s − 1.35·19-s − 0.369·20-s + 0.0997·22-s − 0.400·23-s − 0.453·25-s + 1.22·26-s − 0.455·28-s − 0.910·29-s + 0.150·31-s − 0.176·32-s + 0.673·35-s + 0.191·37-s + 0.957·38-s + 0.261·40-s − 0.631·41-s − 0.404·43-s − 0.0705·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1938798681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1938798681\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 1.65T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 + 0.467T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 0.837T + 31T^{2} \) |
| 37 | \( 1 - 1.16T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 + 2.65T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 + 1.49T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 + 8.88T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
| 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
−8.100559117851632422948629999838, −7.60988231361865416788941983174, −6.89592178187164283919230223330, −6.31078708000706910423593226016, −5.36101042272229066330952217566, −4.42807152942747490017223620274, −3.63907311853210754264449643805, −2.71210938374123299119183805091, −1.93571569825874581389759640337, −0.24643977843565454329746127275,
0.24643977843565454329746127275, 1.93571569825874581389759640337, 2.71210938374123299119183805091, 3.63907311853210754264449643805, 4.42807152942747490017223620274, 5.36101042272229066330952217566, 6.31078708000706910423593226016, 6.89592178187164283919230223330, 7.60988231361865416788941983174, 8.100559117851632422948629999838
