L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 11-s − 2·13-s + 16-s − 17-s + 3·19-s + 2·20-s + 22-s − 23-s − 25-s − 2·26-s − 29-s + 2·31-s + 32-s − 34-s − 5·37-s + 3·38-s + 2·40-s + 10·41-s + 43-s + 44-s − 46-s − 7·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.688·19-s + 0.447·20-s + 0.213·22-s − 0.208·23-s − 1/5·25-s − 0.392·26-s − 0.185·29-s + 0.359·31-s + 0.176·32-s − 0.171·34-s − 0.821·37-s + 0.486·38-s + 0.316·40-s + 1.56·41-s + 0.152·43-s + 0.150·44-s − 0.147·46-s − 1.02·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.217923584\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.217923584\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−7.51382101046354583458382716216, −6.82327095204912700731615228922, −6.27880152875356204985725794452, −5.46726822597050484073731652654, −5.14023744914147607428815783190, −4.16534012595490278482957035776, −3.52574060456582762358760217295, −2.49830912700670098421764211098, −2.01742704906610782591329295724, −0.884800001171135008106598670913,
0.884800001171135008106598670913, 2.01742704906610782591329295724, 2.49830912700670098421764211098, 3.52574060456582762358760217295, 4.16534012595490278482957035776, 5.14023744914147607428815783190, 5.46726822597050484073731652654, 6.27880152875356204985725794452, 6.82327095204912700731615228922, 7.51382101046354583458382716216