L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·11-s + 2·13-s + 14-s + 16-s − 2·19-s + 2·22-s − 23-s − 2·26-s − 28-s − 2·31-s − 32-s + 2·37-s + 2·38-s + 6·41-s − 2·43-s − 2·44-s + 46-s − 2·47-s + 49-s + 2·52-s − 10·53-s + 56-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.458·19-s + 0.426·22-s − 0.208·23-s − 0.392·26-s − 0.188·28-s − 0.359·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s + 0.937·41-s − 0.304·43-s − 0.301·44-s + 0.147·46-s − 0.291·47-s + 1/7·49-s + 0.277·52-s − 1.37·53-s + 0.133·56-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.076270385\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076270385\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.19611294884095, −13.51832901658453, −13.07987135309664, −12.47947440675749, −12.26162679982573, −11.30796652854491, −11.05771776428784, −10.64619199526155, −9.964529152086187, −9.523433575057022, −9.153297859127929, −8.352986746433559, −8.084007386535552, −7.581183728724969, −6.773525583035435, −6.502703693395563, −5.837883854438697, −5.290738654260324, −4.607070892140168, −3.826819922853189, −3.339665165310832, −2.552909496434677, −2.054098623212342, −1.197774342001081, −0.4090808929404745,
0.4090808929404745, 1.197774342001081, 2.054098623212342, 2.552909496434677, 3.339665165310832, 3.826819922853189, 4.607070892140168, 5.290738654260324, 5.837883854438697, 6.502703693395563, 6.773525583035435, 7.581183728724969, 8.084007386535552, 8.352986746433559, 9.153297859127929, 9.523433575057022, 9.964529152086187, 10.64619199526155, 11.05771776428784, 11.30796652854491, 12.26162679982573, 12.47947440675749, 13.07987135309664, 13.51832901658453, 14.19611294884095