L(s) = 1 | + 2.55·3-s − 2.69·5-s + 1.01·7-s + 3.53·9-s − 2.93·11-s − 13-s − 6.88·15-s − 4.17·17-s + 1.18·19-s + 2.58·21-s + 6.85·23-s + 2.26·25-s + 1.36·27-s − 29-s + 4.62·31-s − 7.50·33-s − 2.72·35-s − 4.42·37-s − 2.55·39-s + 5.14·41-s − 3.08·43-s − 9.51·45-s + 9.02·47-s − 5.97·49-s − 10.6·51-s − 6.85·53-s + 7.91·55-s + ⋯ |
L(s) = 1 | + 1.47·3-s − 1.20·5-s + 0.381·7-s + 1.17·9-s − 0.885·11-s − 0.277·13-s − 1.77·15-s − 1.01·17-s + 0.271·19-s + 0.563·21-s + 1.42·23-s + 0.452·25-s + 0.261·27-s − 0.185·29-s + 0.830·31-s − 1.30·33-s − 0.460·35-s − 0.727·37-s − 0.409·39-s + 0.804·41-s − 0.470·43-s − 1.41·45-s + 1.31·47-s − 0.854·49-s − 1.49·51-s − 0.941·53-s + 1.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 - 2.55T + 3T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 17 | \( 1 + 4.17T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 + 3.08T + 43T^{2} \) |
| 47 | \( 1 - 9.02T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 7.80T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 + 1.51T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 7.59T + 79T^{2} \) |
| 83 | \( 1 + 1.23T + 83T^{2} \) |
| 89 | \( 1 - 8.99T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.76815300939254185523574151920, −7.40075612414348714120986984520, −6.61873088444712429502452329258, −5.35319973164817794161698120725, −4.55797604006326488795227079679, −4.00920096882246790783201759284, −3.01441648757327375532802742651, −2.67984053317663685094997810118, −1.51016222367431392740855512969, 0,
1.51016222367431392740855512969, 2.67984053317663685094997810118, 3.01441648757327375532802742651, 4.00920096882246790783201759284, 4.55797604006326488795227079679, 5.35319973164817794161698120725, 6.61873088444712429502452329258, 7.40075612414348714120986984520, 7.76815300939254185523574151920