L(s) = 1 | + 2-s + 0.921·3-s + 4-s − 5-s + 0.921·6-s − 3.84·7-s + 8-s − 2.15·9-s − 10-s + 11-s + 0.921·12-s + 3.36·13-s − 3.84·14-s − 0.921·15-s + 16-s + 4.48·17-s − 2.15·18-s − 6.54·19-s − 20-s − 3.54·21-s + 22-s + 2.73·23-s + 0.921·24-s + 25-s + 3.36·26-s − 4.74·27-s − 3.84·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.532·3-s + 0.5·4-s − 0.447·5-s + 0.376·6-s − 1.45·7-s + 0.353·8-s − 0.716·9-s − 0.316·10-s + 0.301·11-s + 0.266·12-s + 0.932·13-s − 1.02·14-s − 0.237·15-s + 0.250·16-s + 1.08·17-s − 0.506·18-s − 1.50·19-s − 0.223·20-s − 0.772·21-s + 0.213·22-s + 0.571·23-s + 0.188·24-s + 0.200·25-s + 0.659·26-s − 0.913·27-s − 0.726·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.726452196\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.726452196\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 - 0.921T + 3T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 - 4.48T + 17T^{2} \) |
| 19 | \( 1 + 6.54T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 - 2.61T + 29T^{2} \) |
| 31 | \( 1 + 0.588T + 31T^{2} \) |
| 37 | \( 1 + 0.240T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 47 | \( 1 - 9.27T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 9.61T + 61T^{2} \) |
| 67 | \( 1 + 8.11T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.98T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−8.331311083860331222807306405996, −7.49423073815917276473559889617, −6.71021925854088288854341154459, −6.06510276384026228861638993670, −5.55888449043772419480066683671, −4.25373744405142039508612158823, −3.70474681778706273801281176176, −3.06506044231373806997051682747, −2.34521341721808445025243822474, −0.78166653688493349107480652763,
0.78166653688493349107480652763, 2.34521341721808445025243822474, 3.06506044231373806997051682747, 3.70474681778706273801281176176, 4.25373744405142039508612158823, 5.55888449043772419480066683671, 6.06510276384026228861638993670, 6.71021925854088288854341154459, 7.49423073815917276473559889617, 8.331311083860331222807306405996