| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 8·19-s − 20-s + 21-s + 22-s + 24-s + 25-s + 2·26-s + 27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.757952536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.757952536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.75451865622864, −14.29251981034941, −13.69424554975853, −13.51374493213097, −12.68681270205039, −12.29648219841494, −11.72681978449018, −11.15533054800342, −10.89272642022730, −10.01428656993291, −9.435984607048632, −9.032132829754390, −8.228026328895517, −7.748981149041766, −7.334182336606070, −6.750552651883356, −5.950854763902573, −5.408975207337870, −4.845033242127742, −4.114110345598144, −3.494272794155710, −3.225415672627258, −2.260863330490797, −1.551385221525186, −0.7886096265150911,
0.7886096265150911, 1.551385221525186, 2.260863330490797, 3.225415672627258, 3.494272794155710, 4.114110345598144, 4.845033242127742, 5.408975207337870, 5.950854763902573, 6.750552651883356, 7.334182336606070, 7.748981149041766, 8.228026328895517, 9.032132829754390, 9.435984607048632, 10.01428656993291, 10.89272642022730, 11.15533054800342, 11.72681978449018, 12.29648219841494, 12.68681270205039, 13.51374493213097, 13.69424554975853, 14.29251981034941, 14.75451865622864
