| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s + 24·5-s + 36·6-s + 49·7-s + 64·8-s + 81·9-s + 96·10-s + 374·11-s + 144·12-s − 169·13-s + 196·14-s + 216·15-s + 256·16-s + 1.78e3·17-s + 324·18-s + 584·19-s + 384·20-s + 441·21-s + 1.49e3·22-s − 884·23-s + 576·24-s − 2.54e3·25-s − 676·26-s + 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.429·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.303·10-s + 0.931·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.247·15-s + 1/4·16-s + 1.49·17-s + 0.235·18-s + 0.371·19-s + 0.214·20-s + 0.218·21-s + 0.658·22-s − 0.348·23-s + 0.204·24-s − 0.815·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.759412562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.759412562\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 24 T + p^{5} T^{2} \) |
| 11 | \( 1 - 34 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1786 T + p^{5} T^{2} \) |
| 19 | \( 1 - 584 T + p^{5} T^{2} \) |
| 23 | \( 1 + 884 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5878 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6196 T + p^{5} T^{2} \) |
| 37 | \( 1 + 2690 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1848 T + p^{5} T^{2} \) |
| 43 | \( 1 - 22164 T + p^{5} T^{2} \) |
| 47 | \( 1 + 4190 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1082 T + p^{5} T^{2} \) |
| 59 | \( 1 + 14286 T + p^{5} T^{2} \) |
| 61 | \( 1 + 8390 T + p^{5} T^{2} \) |
| 67 | \( 1 + 21508 T + p^{5} T^{2} \) |
| 71 | \( 1 - 19266 T + p^{5} T^{2} \) |
| 73 | \( 1 - 71926 T + p^{5} T^{2} \) |
| 79 | \( 1 + 7184 T + p^{5} T^{2} \) |
| 83 | \( 1 - 99134 T + p^{5} T^{2} \) |
| 89 | \( 1 + 19940 T + p^{5} T^{2} \) |
| 97 | \( 1 + 42322 T + p^{5} T^{2} \) |
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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
−9.885240266892649982737622367307, −9.306131619594005701412150024332, −8.044422650945645202408761741289, −7.37222894944364103571283870721, −6.20081443738320407426612638367, −5.38620051546904874914067333001, −4.21416175571532196737273372813, −3.34592926884514797690263902803, −2.15063478935988729540591161304, −1.14033784222359976608919786923,
1.14033784222359976608919786923, 2.15063478935988729540591161304, 3.34592926884514797690263902803, 4.21416175571532196737273372813, 5.38620051546904874914067333001, 6.20081443738320407426612638367, 7.37222894944364103571283870721, 8.044422650945645202408761741289, 9.306131619594005701412150024332, 9.885240266892649982737622367307
