| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.870996888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.870996888\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 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
−9.547040289033150690294822441027, −8.613628586363707276957269542969, −7.37401646943555492580120617253, −6.73915837939670470281220727226, −5.97447010434826597680072551041, −5.32734843201609780311424178486, −4.47367404653766349933628761316, −3.49143456848613430686623125279, −2.28380910146500603557363711542, −1.19155942635042385588089355891,
1.19155942635042385588089355891, 2.28380910146500603557363711542, 3.49143456848613430686623125279, 4.47367404653766349933628761316, 5.32734843201609780311424178486, 5.97447010434826597680072551041, 6.73915837939670470281220727226, 7.37401646943555492580120617253, 8.613628586363707276957269542969, 9.547040289033150690294822441027
