L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s − 3·9-s + 10-s − 4·11-s − 6·13-s + 14-s + 16-s − 3·18-s + 20-s − 4·22-s + 25-s − 6·26-s + 28-s − 6·29-s − 8·31-s + 32-s + 35-s − 3·36-s + 10·37-s + 40-s − 2·41-s + 4·43-s − 4·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.707·18-s + 0.223·20-s − 0.852·22-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.169·35-s − 1/2·36-s + 1.64·37-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.290126819\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.290126819\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−15.42195748348554, −14.88772161595554, −14.62668953800555, −14.07823800198410, −13.47406577802398, −12.91398675265495, −12.49491617749747, −11.87285739070608, −11.25076273806751, −10.77860804981073, −10.26468595757974, −9.407283349047253, −9.139692296564201, −8.045380018373238, −7.715008298764986, −7.177987781624503, −6.317993225006189, −5.544194312637755, −5.343421336099920, −4.726946151692229, −3.893426186392603, −2.991815011699154, −2.411259539033541, −1.987579564558808, −0.5133476471772252,
0.5133476471772252, 1.987579564558808, 2.411259539033541, 2.991815011699154, 3.893426186392603, 4.726946151692229, 5.343421336099920, 5.544194312637755, 6.317993225006189, 7.177987781624503, 7.715008298764986, 8.045380018373238, 9.139692296564201, 9.407283349047253, 10.26468595757974, 10.77860804981073, 11.25076273806751, 11.87285739070608, 12.49491617749747, 12.91398675265495, 13.47406577802398, 14.07823800198410, 14.62668953800555, 14.88772161595554, 15.42195748348554