L(s) = 1 | − 2-s − 3-s + 4-s − 3.43·5-s + 6-s − 0.875·7-s − 8-s + 9-s + 3.43·10-s − 4.12·11-s − 12-s − 13-s + 0.875·14-s + 3.43·15-s + 16-s + 2.91·17-s − 18-s + 6.27·19-s − 3.43·20-s + 0.875·21-s + 4.12·22-s − 2.22·23-s + 24-s + 6.81·25-s + 26-s − 27-s − 0.875·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.53·5-s + 0.408·6-s − 0.330·7-s − 0.353·8-s + 0.333·9-s + 1.08·10-s − 1.24·11-s − 0.288·12-s − 0.277·13-s + 0.234·14-s + 0.887·15-s + 0.250·16-s + 0.706·17-s − 0.235·18-s + 1.43·19-s − 0.768·20-s + 0.191·21-s + 0.880·22-s − 0.464·23-s + 0.204·24-s + 1.36·25-s + 0.196·26-s − 0.192·27-s − 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2785838900\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2785838900\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 + 0.875T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 31 | \( 1 - 4.78T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 7.01T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 + 0.740T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 - 3.84T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.79723907784136109263029735605, −7.41926217876655982354087006405, −6.66775494634936895240423626894, −5.74567810269351740502956900704, −5.09723994294327699191927821565, −4.31316725681119835651136358612, −3.32763155216261600561771709714, −2.85548544470934900589785544821, −1.42141121113758165444200392351, −0.31285552070410861934162807343,
0.31285552070410861934162807343, 1.42141121113758165444200392351, 2.85548544470934900589785544821, 3.32763155216261600561771709714, 4.31316725681119835651136358612, 5.09723994294327699191927821565, 5.74567810269351740502956900704, 6.66775494634936895240423626894, 7.41926217876655982354087006405, 7.79723907784136109263029735605