| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 4·11-s − 13-s − 15-s + 17-s + 4·21-s + 25-s − 27-s − 6·29-s + 4·31-s + 4·33-s − 4·35-s + 10·37-s + 39-s − 2·41-s + 4·43-s + 45-s + 9·49-s − 51-s + 10·53-s − 4·55-s − 4·59-s + 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s − 0.676·35-s + 1.64·37-s + 0.160·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.140·51-s + 1.37·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511234984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511234984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.99920892696056, −12.71745623428259, −12.12719155444909, −11.68556275711512, −10.96334160042333, −10.72242041679420, −10.11327369044085, −9.723972056105571, −9.504342051467760, −8.913174145089609, −8.118094924720613, −7.813533078142696, −7.087662044490829, −6.805483294965915, −6.160844284925558, −5.784656832391896, −5.393889889455296, −4.806834432327178, −4.166978608930338, −3.561728772739227, −2.989687472633136, −2.451889274935335, −1.979891389218206, −0.8384464777623260, −0.4628629025316463,
0.4628629025316463, 0.8384464777623260, 1.979891389218206, 2.451889274935335, 2.989687472633136, 3.561728772739227, 4.166978608930338, 4.806834432327178, 5.393889889455296, 5.784656832391896, 6.160844284925558, 6.805483294965915, 7.087662044490829, 7.813533078142696, 8.118094924720613, 8.913174145089609, 9.504342051467760, 9.723972056105571, 10.11327369044085, 10.72242041679420, 10.96334160042333, 11.68556275711512, 12.12719155444909, 12.71745623428259, 12.99920892696056
