| L(s) = 1 | − 3.11·3-s + 4.12·5-s + 1.17·7-s + 6.69·9-s − 4.55·11-s − 2.99·13-s − 12.8·15-s − 17-s − 3.45·19-s − 3.65·21-s − 0.461·23-s + 12.0·25-s − 11.5·27-s + 5.51·29-s + 0.588·31-s + 14.1·33-s + 4.83·35-s + 3.02·37-s + 9.32·39-s − 4.05·41-s + 8.04·43-s + 27.6·45-s + 1.26·47-s − 5.62·49-s + 3.11·51-s − 5.96·53-s − 18.7·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 1.84·5-s + 0.443·7-s + 2.23·9-s − 1.37·11-s − 0.830·13-s − 3.31·15-s − 0.242·17-s − 0.793·19-s − 0.796·21-s − 0.0961·23-s + 2.40·25-s − 2.21·27-s + 1.02·29-s + 0.105·31-s + 2.46·33-s + 0.817·35-s + 0.496·37-s + 1.49·39-s − 0.632·41-s + 1.22·43-s + 4.12·45-s + 0.184·47-s − 0.803·49-s + 0.436·51-s − 0.819·53-s − 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 4.12T + 5T^{2} \) |
| 7 | \( 1 - 1.17T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 2.99T + 13T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 + 0.461T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 - 0.588T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 + 5.96T + 53T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 9.06T + 67T^{2} \) |
| 71 | \( 1 + 0.0811T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 0.604T + 79T^{2} \) |
| 83 | \( 1 + 1.82T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 2.19T + 97T^{2} \) |
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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
−7.19818723321063880428284664387, −6.54473165994262621059132680183, −5.96940813596273431951636757588, −5.49258392369651740222919392336, −4.85821770691718262856272601870, −4.51088643362973817940411045704, −2.76116988978392955487799466580, −2.08485233634353434718592044280, −1.19452328392631576136237346938, 0,
1.19452328392631576136237346938, 2.08485233634353434718592044280, 2.76116988978392955487799466580, 4.51088643362973817940411045704, 4.85821770691718262856272601870, 5.49258392369651740222919392336, 5.96940813596273431951636757588, 6.54473165994262621059132680183, 7.19818723321063880428284664387
