| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 4·11-s + 4·13-s − 14-s + 16-s + 4·17-s + 2·19-s − 20-s + 4·22-s + 23-s + 25-s + 4·26-s − 28-s + 10·29-s + 2·31-s + 32-s + 4·34-s + 35-s + 2·37-s + 2·38-s − 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 1.85·29-s + 0.359·31-s + 0.176·32-s + 0.685·34-s + 0.169·35-s + 0.328·37-s + 0.324·38-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.189148240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.189148240\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.05159065010951, −15.57624611449522, −14.90854609967182, −14.36481784567580, −13.95545219394679, −13.30113527790126, −12.81260667579005, −11.99883305218104, −11.77454575570099, −11.28024783718457, −10.36510252127972, −10.01029412469557, −9.178432182384422, −8.486101442388281, −8.058987445918428, −7.064734383812872, −6.714424184824191, −6.036087740821798, −5.454616768224480, −4.537819894684605, −4.008619580570402, −3.316133940703052, −2.835297404656269, −1.503525839716617, −0.8962947115751464,
0.8962947115751464, 1.503525839716617, 2.835297404656269, 3.316133940703052, 4.008619580570402, 4.537819894684605, 5.454616768224480, 6.036087740821798, 6.714424184824191, 7.064734383812872, 8.058987445918428, 8.486101442388281, 9.178432182384422, 10.01029412469557, 10.36510252127972, 11.28024783718457, 11.77454575570099, 11.99883305218104, 12.81260667579005, 13.30113527790126, 13.95545219394679, 14.36481784567580, 14.90854609967182, 15.57624611449522, 16.05159065010951
