| L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·7-s − 0.999i·8-s + 9-s − 1.73i·11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)18-s + (−0.866 + 1.49i)22-s − i·23-s + (−0.866 + 0.499i)28-s + 1.73i·29-s + (0.866 − 0.499i)32-s + (0.499 + 0.866i)36-s + 1.73·37-s + ⋯ |
| L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + i·7-s − 0.999i·8-s + 9-s − 1.73i·11-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)18-s + (−0.866 + 1.49i)22-s − i·23-s + (−0.866 + 0.499i)28-s + 1.73i·29-s + (0.866 − 0.499i)32-s + (0.499 + 0.866i)36-s + 1.73·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7978513914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7978513914\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT - T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.73T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.559314587622905434836722568504, −8.919871072627111089026601847400, −8.353175885508261546442276795679, −7.49138597965873962041646575268, −6.50277314686949515194335874459, −5.76213838251622997730402030452, −4.44715184985955166488361882859, −3.32281008630357227150847568278, −2.49039640507032635732273873908, −1.13550396773616017793949129346,
1.23622711660360511502588434856, 2.31106944723953951983158795480, 4.11814511370719721310756726033, 4.68097393040551552482862211262, 5.95825078169335581832940843414, 6.86453488460322438690075566461, 7.56844278812981384794367257057, 7.79537218923661037734334635009, 9.289521788842409342452276023487, 9.765474911823343634028280596155
