| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 2·11-s + 2·13-s − 15-s + 17-s − 4·19-s + 21-s − 2·23-s + 25-s − 27-s − 2·29-s + 8·31-s − 2·33-s − 35-s − 6·37-s − 2·39-s − 10·41-s + 45-s − 2·47-s + 49-s − 51-s − 10·53-s + 2·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.348·33-s − 0.169·35-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.149·45-s − 0.291·47-s + 1/7·49-s − 0.140·51-s − 1.37·53-s + 0.269·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.442766043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442766043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.60467041807456, −13.19812306286414, −12.55390818397445, −12.13337355487851, −11.80764505494543, −11.10743630819253, −10.72864248586342, −10.20257337432422, −9.768496236616150, −9.322261640115924, −8.599501322033808, −8.339990000978142, −7.622158675563829, −6.915342971331792, −6.423460104273812, −6.258813560197969, −5.588328889489112, −4.968927001383804, −4.493533752679033, −3.764912530616127, −3.351321776648303, −2.546265220566135, −1.775408316121807, −1.314799115359678, −0.3898227150742683,
0.3898227150742683, 1.314799115359678, 1.775408316121807, 2.546265220566135, 3.351321776648303, 3.764912530616127, 4.493533752679033, 4.968927001383804, 5.588328889489112, 6.258813560197969, 6.423460104273812, 6.915342971331792, 7.622158675563829, 8.339990000978142, 8.599501322033808, 9.322261640115924, 9.768496236616150, 10.20257337432422, 10.72864248586342, 11.10743630819253, 11.80764505494543, 12.13337355487851, 12.55390818397445, 13.19812306286414, 13.60467041807456
