L(s) = 1 | − 2-s + 1.21·3-s + 4-s − 5-s − 1.21·6-s − 1.25·7-s − 8-s − 1.51·9-s + 10-s − 6.19·11-s + 1.21·12-s + 13-s + 1.25·14-s − 1.21·15-s + 16-s + 3.94·17-s + 1.51·18-s − 4.85·19-s − 20-s − 1.53·21-s + 6.19·22-s + 8.13·23-s − 1.21·24-s + 25-s − 26-s − 5.50·27-s − 1.25·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.704·3-s + 0.5·4-s − 0.447·5-s − 0.498·6-s − 0.474·7-s − 0.353·8-s − 0.503·9-s + 0.316·10-s − 1.86·11-s + 0.352·12-s + 0.277·13-s + 0.335·14-s − 0.314·15-s + 0.250·16-s + 0.956·17-s + 0.356·18-s − 1.11·19-s − 0.223·20-s − 0.334·21-s + 1.32·22-s + 1.69·23-s − 0.249·24-s + 0.200·25-s − 0.196·26-s − 1.05·27-s − 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9492809077\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9492809077\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 1.21T + 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 17 | \( 1 - 3.94T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 8.13T + 23T^{2} \) |
| 29 | \( 1 - 0.0979T + 29T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 7.70T + 67T^{2} \) |
| 71 | \( 1 + 2.68T + 71T^{2} \) |
| 73 | \( 1 + 0.389T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 - 7.44T + 97T^{2} \) |
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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
−8.480737951906342058215653622317, −7.82863494154161212809758345710, −7.33934862680736209964182165871, −6.35840706220335313281817201904, −5.54119263750002512309547247311, −4.71928200653563199564731549349, −3.37415458604777810592516859172, −2.97176165961672960544206391017, −2.11093688118345606583694039324, −0.56605881424058590629518601348,
0.56605881424058590629518601348, 2.11093688118345606583694039324, 2.97176165961672960544206391017, 3.37415458604777810592516859172, 4.71928200653563199564731549349, 5.54119263750002512309547247311, 6.35840706220335313281817201904, 7.33934862680736209964182165871, 7.82863494154161212809758345710, 8.480737951906342058215653622317