| L(s) = 1 | + 2.24·2-s + 1.80·3-s + 3.02·4-s + 4.04·6-s − 5.11·7-s + 2.30·8-s + 0.248·9-s + 4.18·11-s + 5.45·12-s − 5.65·13-s − 11.4·14-s − 0.884·16-s + 0.983·17-s + 0.556·18-s + 0.188·19-s − 9.22·21-s + 9.38·22-s − 7.24·23-s + 4.15·24-s − 12.6·26-s − 4.95·27-s − 15.5·28-s + 4.85·29-s − 6.91·31-s − 6.59·32-s + 7.54·33-s + 2.20·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.04·3-s + 1.51·4-s + 1.64·6-s − 1.93·7-s + 0.815·8-s + 0.0827·9-s + 1.26·11-s + 1.57·12-s − 1.56·13-s − 3.06·14-s − 0.221·16-s + 0.238·17-s + 0.131·18-s + 0.0432·19-s − 2.01·21-s + 2.00·22-s − 1.51·23-s + 0.848·24-s − 2.48·26-s − 0.954·27-s − 2.92·28-s + 0.901·29-s − 1.24·31-s − 1.16·32-s + 1.31·33-s + 0.378·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 3 | \( 1 - 1.80T + 3T^{2} \) |
| 7 | \( 1 + 5.11T + 7T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 0.983T + 17T^{2} \) |
| 19 | \( 1 - 0.188T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 - 6.75T + 61T^{2} \) |
| 67 | \( 1 - 1.23T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−7.41962074074910285972820418771, −6.81312877578229242142191185842, −6.21578890209166696262110781787, −5.65158619660269940287588946851, −4.56764803727194743586002251185, −3.87157153829378248761877466886, −3.31454326314302296108564550579, −2.76794893186072278501781826450, −2.00873207453967356469126668181, 0,
2.00873207453967356469126668181, 2.76794893186072278501781826450, 3.31454326314302296108564550579, 3.87157153829378248761877466886, 4.56764803727194743586002251185, 5.65158619660269940287588946851, 6.21578890209166696262110781787, 6.81312877578229242142191185842, 7.41962074074910285972820418771
