L(s) = 1 | + 0.692·3-s − 3.58·5-s + 3.81·7-s − 2.52·9-s − 3.20·11-s + 3.92·13-s − 2.48·15-s − 17-s + 0.300·19-s + 2.63·21-s − 1.11·23-s + 7.84·25-s − 3.82·27-s + 7.45·29-s + 0.368·31-s − 2.21·33-s − 13.6·35-s − 5.36·37-s + 2.71·39-s + 0.470·41-s + 8.69·43-s + 9.03·45-s − 8.28·47-s + 7.52·49-s − 0.692·51-s − 14.2·53-s + 11.4·55-s + ⋯ |
L(s) = 1 | + 0.399·3-s − 1.60·5-s + 1.44·7-s − 0.840·9-s − 0.965·11-s + 1.08·13-s − 0.641·15-s − 0.242·17-s + 0.0689·19-s + 0.576·21-s − 0.233·23-s + 1.56·25-s − 0.735·27-s + 1.38·29-s + 0.0661·31-s − 0.386·33-s − 2.30·35-s − 0.882·37-s + 0.435·39-s + 0.0734·41-s + 1.32·43-s + 1.34·45-s − 1.20·47-s + 1.07·49-s − 0.0970·51-s − 1.96·53-s + 1.54·55-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 | \( 1 - 0.692T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 + 3.20T + 11T^{2} \) |
| 13 | \( 1 - 3.92T + 13T^{2} \) |
| 19 | \( 1 - 0.300T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 - 7.45T + 29T^{2} \) |
| 31 | \( 1 - 0.368T + 31T^{2} \) |
| 37 | \( 1 + 5.36T + 37T^{2} \) |
| 41 | \( 1 - 0.470T + 41T^{2} \) |
| 43 | \( 1 - 8.69T + 43T^{2} \) |
| 47 | \( 1 + 8.28T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 61 | \( 1 + 2.58T + 61T^{2} \) |
| 67 | \( 1 + 6.13T + 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 - 0.352T + 73T^{2} \) |
| 79 | \( 1 - 0.179T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−8.109011639820419121419949443774, −7.77909421809421786803340181741, −6.78466811617058959619135918805, −5.74239845866925256254357040559, −4.88552819439562207196391002476, −4.29583037419403156257889047008, −3.40342892521674644137820295354, −2.65424623835679030794815915834, −1.37343934252468851156360781326, 0,
1.37343934252468851156360781326, 2.65424623835679030794815915834, 3.40342892521674644137820295354, 4.29583037419403156257889047008, 4.88552819439562207196391002476, 5.74239845866925256254357040559, 6.78466811617058959619135918805, 7.77909421809421786803340181741, 8.109011639820419121419949443774