| L(s) = 1 | − 0.364·3-s − 2.86·9-s + 1.36·11-s − 2.63·13-s + 2.23·17-s + 6.23·19-s − 6.59·23-s + 2.13·27-s + 5.50·29-s − 4.50·31-s − 0.497·33-s + 2.09·37-s + 0.960·39-s − 9.32·41-s + 1.86·43-s − 6.86·47-s − 0.813·51-s + 10.1·53-s − 2.27·57-s + 1.63·59-s − 0.0394·61-s + 6.72·67-s + 2.40·69-s − 6.27·71-s − 4·73-s + 5.32·79-s + 7.82·81-s + ⋯ |
| L(s) = 1 | − 0.210·3-s − 0.955·9-s + 0.411·11-s − 0.730·13-s + 0.541·17-s + 1.42·19-s − 1.37·23-s + 0.411·27-s + 1.02·29-s − 0.808·31-s − 0.0865·33-s + 0.344·37-s + 0.153·39-s − 1.45·41-s + 0.284·43-s − 1.00·47-s − 0.113·51-s + 1.39·53-s − 0.300·57-s + 0.212·59-s − 0.00505·61-s + 0.822·67-s + 0.289·69-s − 0.744·71-s − 0.468·73-s + 0.599·79-s + 0.869·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.364T + 3T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 + 4.50T + 31T^{2} \) |
| 37 | \( 1 - 2.09T + 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 + 0.0394T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 0.867T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−7.971206688611344046459039780615, −7.16658950971811917340978676884, −6.45599400572719204882499138864, −5.56440975021499183724389494499, −5.19685020958603945136686937170, −4.12354757703870222598903484522, −3.27114606555612804085389851130, −2.49221441059160069332541343509, −1.30534389583557870960338611247, 0,
1.30534389583557870960338611247, 2.49221441059160069332541343509, 3.27114606555612804085389851130, 4.12354757703870222598903484522, 5.19685020958603945136686937170, 5.56440975021499183724389494499, 6.45599400572719204882499138864, 7.16658950971811917340978676884, 7.971206688611344046459039780615
