| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 44·5-s + 36·6-s − 70·7-s − 64·8-s + 81·9-s + 176·10-s − 136·11-s − 144·12-s − 1.02e3·13-s + 280·14-s + 396·15-s + 256·16-s + 484·17-s − 324·18-s − 1.04e3·19-s − 704·20-s + 630·21-s + 544·22-s + 529·23-s + 576·24-s − 1.18e3·25-s + 4.08e3·26-s − 729·27-s − 1.12e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.787·5-s + 0.408·6-s − 0.539·7-s − 0.353·8-s + 1/3·9-s + 0.556·10-s − 0.338·11-s − 0.288·12-s − 1.67·13-s + 0.381·14-s + 0.454·15-s + 1/4·16-s + 0.406·17-s − 0.235·18-s − 0.664·19-s − 0.393·20-s + 0.311·21-s + 0.239·22-s + 0.208·23-s + 0.204·24-s − 0.380·25-s + 1.18·26-s − 0.192·27-s − 0.269·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4922499321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4922499321\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| 23 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 44 T + p^{5} T^{2} \) |
| 7 | \( 1 + 10 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 136 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1022 T + p^{5} T^{2} \) |
| 17 | \( 1 - 484 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1046 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2618 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4860 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14918 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7530 T + p^{5} T^{2} \) |
| 43 | \( 1 - 16186 T + p^{5} T^{2} \) |
| 47 | \( 1 - 29160 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9896 T + p^{5} T^{2} \) |
| 59 | \( 1 + 2004 T + p^{5} T^{2} \) |
| 61 | \( 1 + 2570 T + p^{5} T^{2} \) |
| 67 | \( 1 - 46118 T + p^{5} T^{2} \) |
| 71 | \( 1 + 32688 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46830 T + p^{5} T^{2} \) |
| 79 | \( 1 + 34338 T + p^{5} T^{2} \) |
| 83 | \( 1 - 31736 T + p^{5} T^{2} \) |
| 89 | \( 1 + 60792 T + p^{5} T^{2} \) |
| 97 | \( 1 + 19218 T + p^{5} T^{2} \) |
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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
−12.17779805485385552356051401254, −11.18179690988935330880966761581, −10.18873245663692029326554340626, −9.289391098805120791996858965282, −7.83649902046135058588683531511, −7.13413727329501694723030624812, −5.77647419609062475424271185150, −4.29443218330072935875139841553, −2.57469101954084882867861581150, −0.51101417711952787467331189258,
0.51101417711952787467331189258, 2.57469101954084882867861581150, 4.29443218330072935875139841553, 5.77647419609062475424271185150, 7.13413727329501694723030624812, 7.83649902046135058588683531511, 9.289391098805120791996858965282, 10.18873245663692029326554340626, 11.18179690988935330880966761581, 12.17779805485385552356051401254
