L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 4.64·11-s − 0.645·13-s + 16-s − 3.29·17-s + 4.29·19-s + 20-s + 4.64·22-s + 3·23-s + 25-s − 0.645·26-s − 2·31-s + 32-s − 3.29·34-s − 5.93·37-s + 4.29·38-s + 40-s − 1.35·41-s + 11.2·43-s + 4.64·44-s + 3·46-s + 9.58·47-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.40·11-s − 0.179·13-s + 0.250·16-s − 0.798·17-s + 0.984·19-s + 0.223·20-s + 0.990·22-s + 0.625·23-s + 0.200·25-s − 0.126·26-s − 0.359·31-s + 0.176·32-s − 0.564·34-s − 0.976·37-s + 0.696·38-s + 0.158·40-s − 0.211·41-s + 1.72·43-s + 0.700·44-s + 0.442·46-s + 1.39·47-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.832726777\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.832726777\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 0.645T + 13T^{2} \) |
| 17 | \( 1 + 3.29T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 9.58T + 47T^{2} \) |
| 53 | \( 1 + 0.291T + 53T^{2} \) |
| 59 | \( 1 - 9.29T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 + 2.70T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−8.425846434236206497863082466508, −7.23636938022611635180405181980, −6.95462230478987972238319822198, −6.04607605289126903572218879953, −5.46587000699864891797960677019, −4.56231306648423823216616106016, −3.88415136559856612535616225579, −3.00774009391432335650527508599, −2.04074970278051878996166009478, −1.05595286642043370925709543825,
1.05595286642043370925709543825, 2.04074970278051878996166009478, 3.00774009391432335650527508599, 3.88415136559856612535616225579, 4.56231306648423823216616106016, 5.46587000699864891797960677019, 6.04607605289126903572218879953, 6.95462230478987972238319822198, 7.23636938022611635180405181980, 8.425846434236206497863082466508