| L(s) = 1 | + 2·2-s + 5·3-s + 4·4-s + 3·5-s + 10·6-s − 32·7-s + 8·8-s − 2·9-s + 6·10-s + 4·11-s + 20·12-s − 69·13-s − 64·14-s + 15·15-s + 16·16-s + 19·17-s − 4·18-s + 12·20-s − 160·21-s + 8·22-s + 67·23-s + 40·24-s − 116·25-s − 138·26-s − 145·27-s − 128·28-s + 51·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.962·3-s + 1/2·4-s + 0.268·5-s + 0.680·6-s − 1.72·7-s + 0.353·8-s − 0.0740·9-s + 0.189·10-s + 0.109·11-s + 0.481·12-s − 1.47·13-s − 1.22·14-s + 0.258·15-s + 1/4·16-s + 0.271·17-s − 0.0523·18-s + 0.134·20-s − 1.66·21-s + 0.0775·22-s + 0.607·23-s + 0.340·24-s − 0.927·25-s − 1.04·26-s − 1.03·27-s − 0.863·28-s + 0.326·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 69 T + p^{3} T^{2} \) |
| 17 | \( 1 - 19 T + p^{3} T^{2} \) |
| 23 | \( 1 - 67 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 + 132 T + p^{3} T^{2} \) |
| 37 | \( 1 + 14 T + p^{3} T^{2} \) |
| 41 | \( 1 + 413 T + p^{3} T^{2} \) |
| 43 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 617 T + p^{3} T^{2} \) |
| 53 | \( 1 - 383 T + p^{3} T^{2} \) |
| 59 | \( 1 + 599 T + p^{3} T^{2} \) |
| 61 | \( 1 + 217 T + p^{3} T^{2} \) |
| 67 | \( 1 + 225 T + p^{3} T^{2} \) |
| 71 | \( 1 - 701 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1015 T + p^{3} T^{2} \) |
| 79 | \( 1 - 349 T + p^{3} T^{2} \) |
| 83 | \( 1 + 592 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1349 T + p^{3} T^{2} \) |
| 97 | \( 1 + 613 T + p^{3} T^{2} \) |
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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
−9.693782560393120798689203731269, −8.863724102244458706468742891104, −7.68842273798202201971663994799, −6.89302305059085453243864357737, −6.02853472728559873145774136137, −5.00540227975715430459249984068, −3.62829091706105748067338853938, −3.03355697586594280891801641983, −2.12350705464138638605025404901, 0,
2.12350705464138638605025404901, 3.03355697586594280891801641983, 3.62829091706105748067338853938, 5.00540227975715430459249984068, 6.02853472728559873145774136137, 6.89302305059085453243864357737, 7.68842273798202201971663994799, 8.863724102244458706468742891104, 9.693782560393120798689203731269
