| L(s) = 1 | + 2·2-s + 3-s + 4·4-s − 5·5-s + 2·6-s + 8·8-s − 26·9-s − 10·10-s − 9·11-s + 4·12-s + 51·13-s − 5·15-s + 16·16-s + 81·17-s − 52·18-s + 86·19-s − 20·20-s − 18·22-s + 48·23-s + 8·24-s + 25·25-s + 102·26-s − 53·27-s + 211·29-s − 10·30-s + 254·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s + 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.246·11-s + 0.0962·12-s + 1.08·13-s − 0.0860·15-s + 1/4·16-s + 1.15·17-s − 0.680·18-s + 1.03·19-s − 0.223·20-s − 0.174·22-s + 0.435·23-s + 0.0680·24-s + 1/5·25-s + 0.769·26-s − 0.377·27-s + 1.35·29-s − 0.0608·30-s + 1.47·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.109871364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109871364\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 51 T + p^{3} T^{2} \) |
| 17 | \( 1 - 81 T + p^{3} T^{2} \) |
| 19 | \( 1 - 86 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 211 T + p^{3} T^{2} \) |
| 31 | \( 1 - 254 T + p^{3} T^{2} \) |
| 37 | \( 1 + 20 T + p^{3} T^{2} \) |
| 41 | \( 1 - 74 T + p^{3} T^{2} \) |
| 43 | \( 1 + 318 T + p^{3} T^{2} \) |
| 47 | \( 1 + 167 T + p^{3} T^{2} \) |
| 53 | \( 1 + 170 T + p^{3} T^{2} \) |
| 59 | \( 1 - 854 T + p^{3} T^{2} \) |
| 61 | \( 1 + 580 T + p^{3} T^{2} \) |
| 67 | \( 1 + 58 T + p^{3} T^{2} \) |
| 71 | \( 1 - 152 T + p^{3} T^{2} \) |
| 73 | \( 1 - 702 T + p^{3} T^{2} \) |
| 79 | \( 1 + 419 T + p^{3} T^{2} \) |
| 83 | \( 1 - 124 T + p^{3} T^{2} \) |
| 89 | \( 1 + 768 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1085 T + p^{3} T^{2} \) |
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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
−10.74544660592948715348526179487, −9.764343784036142101631317643615, −8.480589370949020232345106614188, −7.943496589945695576612314717470, −6.72052700670370273962164667558, −5.76446327819599086762148416343, −4.86174605100795712244098504845, −3.52653464909152759575468590866, −2.85618348169017767862010679183, −1.03637908679416231338013220445,
1.03637908679416231338013220445, 2.85618348169017767862010679183, 3.52653464909152759575468590866, 4.86174605100795712244098504845, 5.76446327819599086762148416343, 6.72052700670370273962164667558, 7.943496589945695576612314717470, 8.480589370949020232345106614188, 9.764343784036142101631317643615, 10.74544660592948715348526179487
