L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 4·11-s + 6·13-s − 2·15-s + 17-s + 21-s − 25-s + 27-s + 2·29-s − 4·33-s − 2·35-s + 6·37-s + 6·39-s + 2·41-s + 4·43-s − 2·45-s + 49-s + 51-s − 2·53-s + 8·55-s − 4·59-s + 2·61-s + 63-s − 12·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.516·15-s + 0.242·17-s + 0.218·21-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.696·33-s − 0.338·35-s + 0.986·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.140·51-s − 0.274·53-s + 1.07·55-s − 0.520·59-s + 0.256·61-s + 0.125·63-s − 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.989630681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.989630681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 14 T + p T^{2} \) |
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
−8.588533969061071524284588272560, −7.915035693401358032291429796467, −7.70501770080330177994289170781, −6.55284452749473279394727010866, −5.70746088776540014205862408741, −4.74348244395772540437944707253, −3.90931343559427893000267483160, −3.23157751360505492878849183085, −2.18394974322044895912107055002, −0.860660161553665057885638838821,
0.860660161553665057885638838821, 2.18394974322044895912107055002, 3.23157751360505492878849183085, 3.90931343559427893000267483160, 4.74348244395772540437944707253, 5.70746088776540014205862408741, 6.55284452749473279394727010866, 7.70501770080330177994289170781, 7.915035693401358032291429796467, 8.588533969061071524284588272560