L(s) = 1 | − 2-s − 2.73·3-s + 4-s + 5-s + 2.73·6-s + 7-s − 8-s + 4.50·9-s − 10-s − 2.73·12-s − 2.65·13-s − 14-s − 2.73·15-s + 16-s + 6.67·17-s − 4.50·18-s − 5.92·19-s + 20-s − 2.73·21-s − 6.62·23-s + 2.73·24-s + 25-s + 2.65·26-s − 4.11·27-s + 28-s − 1.10·29-s + 2.73·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s + 0.377·7-s − 0.353·8-s + 1.50·9-s − 0.316·10-s − 0.790·12-s − 0.735·13-s − 0.267·14-s − 0.707·15-s + 0.250·16-s + 1.61·17-s − 1.06·18-s − 1.35·19-s + 0.223·20-s − 0.597·21-s − 1.38·23-s + 0.559·24-s + 0.200·25-s + 0.520·26-s − 0.792·27-s + 0.188·28-s − 0.205·29-s + 0.500·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5851091707\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5851091707\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 5.92T + 19T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 - 0.907T + 47T^{2} \) |
| 53 | \( 1 + 9.24T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 + 4.10T + 67T^{2} \) |
| 71 | \( 1 + 7.02T + 71T^{2} \) |
| 73 | \( 1 + 0.967T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.67738268947535476383390003834, −7.11716294609594458468962291379, −6.21833535862680983329576642915, −5.90933376010537387951413001717, −5.19896248821186772460049853494, −4.52718939781328673335301423886, −3.53661381111727589123602153766, −2.25128289907222724612791884994, −1.52969848682209069355792916947, −0.46499380803944620236475398123,
0.46499380803944620236475398123, 1.52969848682209069355792916947, 2.25128289907222724612791884994, 3.53661381111727589123602153766, 4.52718939781328673335301423886, 5.19896248821186772460049853494, 5.90933376010537387951413001717, 6.21833535862680983329576642915, 7.11716294609594458468962291379, 7.67738268947535476383390003834