L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 3·11-s + 12-s − 13-s + 14-s + 16-s + 4·17-s + 18-s + 21-s + 3·22-s + 3·23-s + 24-s − 26-s + 27-s + 28-s + 8·29-s + 3·31-s + 32-s + 3·33-s + 4·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.218·21-s + 0.639·22-s + 0.625·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.48·29-s + 0.538·31-s + 0.176·32-s + 0.522·33-s + 0.685·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.540028615\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.540028615\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−12.27175004226077, −12.11693927932729, −11.85164649795774, −11.15796412413655, −10.75040547692290, −10.14488561711458, −9.911205113734631, −9.238656384563673, −8.844284051209497, −8.354635449185616, −7.853468621067151, −7.405819237786499, −6.998439462189238, −6.335208023127155, −6.104817712524069, −5.420714860790512, −4.722434384330715, −4.622528362250650, −3.966779578902732, −3.370984516891639, −2.948214149191937, −2.534626869662175, −1.649102997848821, −1.342444622347215, −0.6422843328830027,
0.6422843328830027, 1.342444622347215, 1.649102997848821, 2.534626869662175, 2.948214149191937, 3.370984516891639, 3.966779578902732, 4.622528362250650, 4.722434384330715, 5.420714860790512, 6.104817712524069, 6.335208023127155, 6.998439462189238, 7.405819237786499, 7.853468621067151, 8.354635449185616, 8.844284051209497, 9.238656384563673, 9.911205113734631, 10.14488561711458, 10.75040547692290, 11.15796412413655, 11.85164649795774, 12.11693927932729, 12.27175004226077