| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s + 4·13-s − 14-s + 16-s − 17-s + 4·19-s + 3·20-s + 6·23-s + 4·25-s + 4·26-s − 28-s + 5·31-s + 32-s − 34-s − 3·35-s + 2·37-s + 4·38-s + 3·40-s + 6·41-s + 43-s + 6·46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.670·20-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 0.188·28-s + 0.898·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s + 0.328·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.152·43-s + 0.884·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.988795666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.988795666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.01335321964633, −12.55057812524368, −11.91605031881290, −11.44092569327026, −11.01233850623080, −10.57601626024558, −10.03742263313685, −9.582228329993030, −9.273850593972846, −8.635291464853022, −8.214595265382993, −7.536453316181418, −6.828318737110944, −6.666556495841827, −6.103119417918711, −5.500081256035762, −5.423401688319214, −4.672035740618739, −4.096397396627630, −3.493464375474472, −2.976101286302024, −2.467121436110521, −1.927386885559606, −1.123687656246314, −0.8069322053237958,
0.8069322053237958, 1.123687656246314, 1.927386885559606, 2.467121436110521, 2.976101286302024, 3.493464375474472, 4.096397396627630, 4.672035740618739, 5.423401688319214, 5.500081256035762, 6.103119417918711, 6.666556495841827, 6.828318737110944, 7.536453316181418, 8.214595265382993, 8.635291464853022, 9.273850593972846, 9.582228329993030, 10.03742263313685, 10.57601626024558, 11.01233850623080, 11.44092569327026, 11.91605031881290, 12.55057812524368, 13.01335321964633
