| L(s) = 1 | − 2.14·2-s + 0.668·3-s + 2.58·4-s − 1.43·6-s + 3.26·7-s − 1.25·8-s − 2.55·9-s + 4.95·11-s + 1.72·12-s + 1.16·13-s − 6.99·14-s − 2.48·16-s + 4.82·17-s + 5.46·18-s − 5.79·19-s + 2.18·21-s − 10.6·22-s − 1.06·23-s − 0.839·24-s − 2.49·26-s − 3.71·27-s + 8.44·28-s − 0.384·29-s − 3.94·31-s + 7.82·32-s + 3.31·33-s − 10.3·34-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.385·3-s + 1.29·4-s − 0.584·6-s + 1.23·7-s − 0.444·8-s − 0.851·9-s + 1.49·11-s + 0.498·12-s + 0.322·13-s − 1.86·14-s − 0.620·16-s + 1.17·17-s + 1.28·18-s − 1.32·19-s + 0.476·21-s − 2.26·22-s − 0.221·23-s − 0.171·24-s − 0.488·26-s − 0.714·27-s + 1.59·28-s − 0.0714·29-s − 0.709·31-s + 1.38·32-s + 0.576·33-s − 1.77·34-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 | \( 1 + 2.14T + 2T^{2} \) |
| 3 | \( 1 - 0.668T + 3T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 0.384T + 29T^{2} \) |
| 31 | \( 1 + 3.94T + 31T^{2} \) |
| 37 | \( 1 - 1.60T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 2.62T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 - 2.79T + 67T^{2} \) |
| 71 | \( 1 + 9.85T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 8.83T + 97T^{2} \) |
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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
−8.121526626860365175273033479969, −7.28882284657079336639315364090, −6.56793760819765008307455900358, −5.79850836273730092438002172596, −4.80496855883138044731943436255, −3.94590117741438349107088629046, −3.00367434496701766385592235432, −1.67512034700120777553602965382, −1.53534885353345990287248969977, 0,
1.53534885353345990287248969977, 1.67512034700120777553602965382, 3.00367434496701766385592235432, 3.94590117741438349107088629046, 4.80496855883138044731943436255, 5.79850836273730092438002172596, 6.56793760819765008307455900358, 7.28882284657079336639315364090, 8.121526626860365175273033479969
