L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 3·11-s − 14-s + 16-s + 6·17-s + 5·19-s + 20-s − 3·22-s + 25-s − 28-s − 4·31-s + 32-s + 6·34-s − 35-s + 11·37-s + 5·38-s + 40-s − 6·41-s + 2·43-s − 3·44-s + 3·47-s − 6·49-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.14·19-s + 0.223·20-s − 0.639·22-s + 1/5·25-s − 0.188·28-s − 0.718·31-s + 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.80·37-s + 0.811·38-s + 0.158·40-s − 0.937·41-s + 0.304·43-s − 0.452·44-s + 0.437·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.853527787\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.853527787\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−16.07428609941936, −15.39465182029802, −14.88034252711243, −14.27883086698053, −13.84053532847809, −13.15835774632534, −12.87312778741435, −12.20000357724200, −11.62928751503201, −11.07217662558596, −10.18806050239044, −10.01001311453326, −9.321085118078369, −8.501768091510699, −7.661119887583280, −7.419829040884092, −6.560615856414053, −5.728187706795610, −5.531071524500611, −4.805530765421424, −3.934583782200110, −3.142445901889181, −2.735372393743836, −1.727332550132964, −0.7743178674447364,
0.7743178674447364, 1.727332550132964, 2.735372393743836, 3.142445901889181, 3.934583782200110, 4.805530765421424, 5.531071524500611, 5.728187706795610, 6.560615856414053, 7.419829040884092, 7.661119887583280, 8.501768091510699, 9.321085118078369, 10.01001311453326, 10.18806050239044, 11.07217662558596, 11.62928751503201, 12.20000357724200, 12.87312778741435, 13.15835774632534, 13.84053532847809, 14.27883086698053, 14.88034252711243, 15.39465182029802, 16.07428609941936