| L(s) = 1 | − 3-s + 4·5-s + 4·7-s + 9-s − 11-s − 13-s − 4·15-s + 4·17-s + 4·19-s − 4·21-s − 8·23-s + 11·25-s − 27-s + 6·31-s + 33-s + 16·35-s − 10·37-s + 39-s − 2·41-s + 10·43-s + 4·45-s − 12·47-s + 9·49-s − 4·51-s − 2·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 1.03·15-s + 0.970·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 11/5·25-s − 0.192·27-s + 1.07·31-s + 0.174·33-s + 2.70·35-s − 1.64·37-s + 0.160·39-s − 0.312·41-s + 1.52·43-s + 0.596·45-s − 1.75·47-s + 9/7·49-s − 0.560·51-s − 0.274·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.136250393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.136250393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.965554062980245718375004016203, −7.26148297476873941862066175090, −6.33346460123499179590248778538, −5.78165492388433725482304274551, −5.09912812695965771022747402380, −4.87027375507636372123179042701, −3.59280612275757784373603652628, −2.37929360478376437739519168035, −1.78500639635310664989774628381, −1.01168014907825126334398878741,
1.01168014907825126334398878741, 1.78500639635310664989774628381, 2.37929360478376437739519168035, 3.59280612275757784373603652628, 4.87027375507636372123179042701, 5.09912812695965771022747402380, 5.78165492388433725482304274551, 6.33346460123499179590248778538, 7.26148297476873941862066175090, 7.965554062980245718375004016203
