L(s) = 1 | − 2.12·2-s + 2.51·4-s − 0.484·5-s − 7-s − 1.09·8-s + 1.03·10-s − 5.60·13-s + 2.12·14-s − 2.70·16-s + 5.60·17-s + 5.28·19-s − 1.21·20-s − 2.48·23-s − 4.76·25-s + 11.9·26-s − 2.51·28-s + 5.28·29-s − 7.12·31-s + 7.93·32-s − 11.9·34-s + 0.484·35-s − 0.235·37-s − 11.2·38-s + 0.530·40-s + 2.39·41-s − 1.03·43-s + 5.28·46-s + ⋯ |
L(s) = 1 | − 1.50·2-s + 1.25·4-s − 0.216·5-s − 0.377·7-s − 0.387·8-s + 0.325·10-s − 1.55·13-s + 0.567·14-s − 0.676·16-s + 1.36·17-s + 1.21·19-s − 0.272·20-s − 0.518·23-s − 0.952·25-s + 2.33·26-s − 0.475·28-s + 0.980·29-s − 1.27·31-s + 1.40·32-s − 2.04·34-s + 0.0819·35-s − 0.0386·37-s − 1.82·38-s + 0.0839·40-s + 0.373·41-s − 0.157·43-s + 0.778·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5532540827\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5532540827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 5 | \( 1 + 0.484T + 5T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 0.235T + 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 - 3.03T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 2.39T + 73T^{2} \) |
| 79 | \( 1 + 9.03T + 79T^{2} \) |
| 83 | \( 1 + 3.21T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 + 8.79T + 97T^{2} \) |
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
−7.80051947495501356004631111890, −7.44044066038661582990948433565, −6.94436913731595214760748151708, −5.87336571580950053796768442086, −5.22634774908598370828258254780, −4.27472614353970426702859079659, −3.27726852256310698547683487435, −2.47145428461977226798772875841, −1.50746858203711461827986552420, −0.48202598319545410468036478655,
0.48202598319545410468036478655, 1.50746858203711461827986552420, 2.47145428461977226798772875841, 3.27726852256310698547683487435, 4.27472614353970426702859079659, 5.22634774908598370828258254780, 5.87336571580950053796768442086, 6.94436913731595214760748151708, 7.44044066038661582990948433565, 7.80051947495501356004631111890