L(s) = 1 | − 3.23·3-s + 0.381·5-s − 3·7-s + 7.47·9-s + 1.61·11-s − 3.61·13-s − 1.23·15-s − 17-s + 2.23·19-s + 9.70·21-s + 2.38·23-s − 4.85·25-s − 14.4·27-s + 5.85·29-s − 9·31-s − 5.23·33-s − 1.14·35-s + 3·37-s + 11.7·39-s + 9·41-s − 1.38·43-s + 2.85·45-s + 3.85·47-s + 2·49-s + 3.23·51-s + 4.70·53-s + 0.618·55-s + ⋯ |
L(s) = 1 | − 1.86·3-s + 0.170·5-s − 1.13·7-s + 2.49·9-s + 0.487·11-s − 1.00·13-s − 0.319·15-s − 0.242·17-s + 0.512·19-s + 2.11·21-s + 0.496·23-s − 0.970·25-s − 2.78·27-s + 1.08·29-s − 1.61·31-s − 0.911·33-s − 0.193·35-s + 0.493·37-s + 1.87·39-s + 1.40·41-s − 0.210·43-s + 0.425·45-s + 0.562·47-s + 0.285·49-s + 0.453·51-s + 0.646·53-s + 0.0833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 1.38T + 43T^{2} \) |
| 47 | \( 1 - 3.85T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 9.18T + 73T^{2} \) |
| 79 | \( 1 - 0.0901T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−7.74880174501418480416351622482, −7.02141969021477093672704730020, −6.57216044732979714779857611362, −5.82857253152736941345570098221, −5.30079120533470527014894192282, −4.44207119154618631833142446474, −3.64797509487096491072916419892, −2.34762787278381892146636415303, −1.01168280477197151417615062610, 0,
1.01168280477197151417615062610, 2.34762787278381892146636415303, 3.64797509487096491072916419892, 4.44207119154618631833142446474, 5.30079120533470527014894192282, 5.82857253152736941345570098221, 6.57216044732979714779857611362, 7.02141969021477093672704730020, 7.74880174501418480416351622482