| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 4·11-s + 12-s − 2·13-s + 15-s + 16-s − 18-s + 20-s − 4·22-s + 4·23-s − 24-s + 25-s + 2·26-s + 27-s − 2·29-s − 30-s − 32-s + 4·33-s + 36-s + 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.223·20-s − 0.852·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.182·30-s − 0.176·32-s + 0.696·33-s + 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.508600389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.508600389\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−12.51356987625018, −12.01240560292078, −11.49257408496487, −11.11287990555208, −10.63658862287236, −10.06864628461993, −9.595709347329224, −9.414400915485017, −8.845049058919932, −8.606023086269008, −7.940103048412839, −7.447378703328792, −7.087560537646677, −6.621388148214030, −6.044044751104179, −5.729913471565537, −4.828370583530943, −4.577704903863898, −3.816120213940782, −3.327928433636184, −2.807537616514430, −2.168495367898725, −1.761160448011103, −1.078465533136113, −0.5689598433870524,
0.5689598433870524, 1.078465533136113, 1.761160448011103, 2.168495367898725, 2.807537616514430, 3.327928433636184, 3.816120213940782, 4.577704903863898, 4.828370583530943, 5.729913471565537, 6.044044751104179, 6.621388148214030, 7.087560537646677, 7.447378703328792, 7.940103048412839, 8.606023086269008, 8.845049058919932, 9.414400915485017, 9.595709347329224, 10.06864628461993, 10.63658862287236, 11.11287990555208, 11.49257408496487, 12.01240560292078, 12.51356987625018
