| L(s) = 1 | + 9·3-s + 80·5-s + 36·7-s + 81·9-s − 36·11-s + 404·13-s + 720·15-s − 2·17-s + 2.91e3·19-s + 324·21-s − 2.80e3·23-s + 3.27e3·25-s + 729·27-s − 4.40e3·29-s + 5.79e3·31-s − 324·33-s + 2.88e3·35-s + 2.31e3·37-s + 3.63e3·39-s − 6.87e3·41-s + 1.72e4·43-s + 6.48e3·45-s − 5.68e3·47-s − 1.55e4·49-s − 18·51-s + 3.20e4·53-s − 2.88e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.43·5-s + 0.277·7-s + 1/3·9-s − 0.0897·11-s + 0.663·13-s + 0.826·15-s − 0.00167·17-s + 1.85·19-s + 0.160·21-s − 1.10·23-s + 1.04·25-s + 0.192·27-s − 0.973·29-s + 1.08·31-s − 0.0517·33-s + 0.397·35-s + 0.278·37-s + 0.382·39-s − 0.638·41-s + 1.42·43-s + 0.477·45-s − 0.375·47-s − 0.922·49-s − 0.000969·51-s + 1.56·53-s − 0.128·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.539490054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.539490054\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 16 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 36 T + p^{5} T^{2} \) |
| 11 | \( 1 + 36 T + p^{5} T^{2} \) |
| 13 | \( 1 - 404 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2916 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2808 T + p^{5} T^{2} \) |
| 29 | \( 1 + 152 p T + p^{5} T^{2} \) |
| 31 | \( 1 - 5796 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2316 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6874 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17244 T + p^{5} T^{2} \) |
| 47 | \( 1 + 5688 T + p^{5} T^{2} \) |
| 53 | \( 1 - 32072 T + p^{5} T^{2} \) |
| 59 | \( 1 - 43308 T + p^{5} T^{2} \) |
| 61 | \( 1 - 18012 T + p^{5} T^{2} \) |
| 67 | \( 1 + 11628 T + p^{5} T^{2} \) |
| 71 | \( 1 + 8712 T + p^{5} T^{2} \) |
| 73 | \( 1 + 15846 T + p^{5} T^{2} \) |
| 79 | \( 1 - 40356 T + p^{5} T^{2} \) |
| 83 | \( 1 + 66204 T + p^{5} T^{2} \) |
| 89 | \( 1 + 66086 T + p^{5} T^{2} \) |
| 97 | \( 1 + 119682 T + p^{5} T^{2} \) |
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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
−9.733589439930003290728986219002, −8.796308294126639102663832814742, −7.947546225354189377143592823151, −6.97394703329251197255913140189, −5.90572959674804223549429111977, −5.31785121745510552599935625630, −4.00834867592225058675066273815, −2.86995432494994893102931971204, −1.91873907922376047954405247988, −1.02479570263085734223655573601,
1.02479570263085734223655573601, 1.91873907922376047954405247988, 2.86995432494994893102931971204, 4.00834867592225058675066273815, 5.31785121745510552599935625630, 5.90572959674804223549429111977, 6.97394703329251197255913140189, 7.947546225354189377143592823151, 8.796308294126639102663832814742, 9.733589439930003290728986219002
