L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 2·15-s + 16-s + 18-s − 4·19-s − 2·20-s + 4·22-s + 24-s − 25-s + 2·26-s + 27-s + 10·29-s − 2·30-s + 8·31-s + 32-s + 4·33-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.85·29-s − 0.365·30-s + 1.43·31-s + 0.176·32-s + 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.828629289\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.828629289\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.90863924309647, −13.67655601311542, −12.89008679367159, −12.38086475226703, −12.06498017392842, −11.65132445676306, −10.94398336778472, −10.65719060402320, −9.998475149580949, −9.243729018830850, −8.920223135204251, −8.272352463621827, −7.854137907549374, −7.359222549679039, −6.660750553517389, −6.194055808458667, −5.896869184589735, −4.671218262246038, −4.424201076442165, −4.043399680362480, −3.395932834902514, −2.806615749901757, −2.222506145986042, −1.277176645512907, −0.7284697158047609,
0.7284697158047609, 1.277176645512907, 2.222506145986042, 2.806615749901757, 3.395932834902514, 4.043399680362480, 4.424201076442165, 4.671218262246038, 5.896869184589735, 6.194055808458667, 6.660750553517389, 7.359222549679039, 7.854137907549374, 8.272352463621827, 8.920223135204251, 9.243729018830850, 9.998475149580949, 10.65719060402320, 10.94398336778472, 11.65132445676306, 12.06498017392842, 12.38086475226703, 12.89008679367159, 13.67655601311542, 13.90863924309647