L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 2·11-s − 12-s + 4·13-s + 2·14-s + 15-s + 16-s + 18-s − 19-s − 20-s − 2·21-s − 2·22-s − 4·23-s − 24-s + 25-s + 4·26-s − 27-s + 2·28-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.436·21-s − 0.426·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.716429319\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.716429319\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.07421449153427, −12.94394728672569, −12.17552762383348, −11.85277705289809, −11.43148930925044, −10.94541968912893, −10.68218996753890, −10.02088271454960, −9.653900519179718, −8.632593021558140, −8.405935482966402, −7.882565030050430, −7.461249338833476, −6.724439378585164, −6.292909138340590, −5.926602095430431, −5.202585847920424, −4.790470622004876, −4.441125953566818, −3.681650668206620, −3.335827972112327, −2.540278853008641, −1.807697776016022, −1.319403002474017, −0.4432241148303816,
0.4432241148303816, 1.319403002474017, 1.807697776016022, 2.540278853008641, 3.335827972112327, 3.681650668206620, 4.441125953566818, 4.790470622004876, 5.202585847920424, 5.926602095430431, 6.292909138340590, 6.724439378585164, 7.461249338833476, 7.882565030050430, 8.405935482966402, 8.632593021558140, 9.653900519179718, 10.02088271454960, 10.68218996753890, 10.94541968912893, 11.43148930925044, 11.85277705289809, 12.17552762383348, 12.94394728672569, 13.07421449153427