| L(s) = 1 | + 9·3-s + 68·5-s + 81·9-s − 388·11-s − 316·13-s + 612·15-s − 1.05e3·17-s + 1.05e3·19-s + 624·23-s + 1.49e3·25-s + 729·27-s + 7.25e3·29-s + 2.29e3·31-s − 3.49e3·33-s + 1.24e4·37-s − 2.84e3·39-s + 5.37e3·41-s + 1.41e4·43-s + 5.50e3·45-s + 4.71e3·47-s − 9.50e3·51-s + 3.78e3·53-s − 2.63e4·55-s + 9.46e3·57-s + 2.52e4·59-s + 2.06e4·61-s − 2.14e4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.21·5-s + 1/3·9-s − 0.966·11-s − 0.518·13-s + 0.702·15-s − 0.886·17-s + 0.668·19-s + 0.245·23-s + 0.479·25-s + 0.192·27-s + 1.60·29-s + 0.429·31-s − 0.558·33-s + 1.49·37-s − 0.299·39-s + 0.499·41-s + 1.16·43-s + 0.405·45-s + 0.311·47-s − 0.511·51-s + 0.184·53-s − 1.17·55-s + 0.385·57-s + 0.944·59-s + 0.711·61-s − 0.630·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.415620941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.415620941\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 68 T + p^{5} T^{2} \) |
| 11 | \( 1 + 388 T + p^{5} T^{2} \) |
| 13 | \( 1 + 316 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1056 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1052 T + p^{5} T^{2} \) |
| 23 | \( 1 - 624 T + p^{5} T^{2} \) |
| 29 | \( 1 - 250 p T + p^{5} T^{2} \) |
| 31 | \( 1 - 2296 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12426 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5376 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14164 T + p^{5} T^{2} \) |
| 47 | \( 1 - 4712 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3782 T + p^{5} T^{2} \) |
| 59 | \( 1 - 25244 T + p^{5} T^{2} \) |
| 61 | \( 1 - 20668 T + p^{5} T^{2} \) |
| 67 | \( 1 - 49012 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4760 T + p^{5} T^{2} \) |
| 73 | \( 1 + 65264 T + p^{5} T^{2} \) |
| 79 | \( 1 + 49736 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7788 T + p^{5} T^{2} \) |
| 89 | \( 1 - 36904 T + p^{5} T^{2} \) |
| 97 | \( 1 + 98264 T + p^{5} T^{2} \) |
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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
−9.883203881948716071212185376368, −9.152077501429281645243656159315, −8.222628152773312766498848566875, −7.28134110646813895995561651222, −6.27434633640315012531541068417, −5.32830885747889095265623134684, −4.37949708278595589571428344960, −2.78095965423076815420331806095, −2.28015138164556957033904331730, −0.881586800319647859429035484990,
0.881586800319647859429035484990, 2.28015138164556957033904331730, 2.78095965423076815420331806095, 4.37949708278595589571428344960, 5.32830885747889095265623134684, 6.27434633640315012531541068417, 7.28134110646813895995561651222, 8.222628152773312766498848566875, 9.152077501429281645243656159315, 9.883203881948716071212185376368
