| L(s) = 1 | − 2.26·2-s + 2.92·3-s + 3.12·4-s + 0.680·5-s − 6.63·6-s − 3.22·7-s − 2.54·8-s + 5.58·9-s − 1.53·10-s + 3.31·11-s + 9.15·12-s − 13-s + 7.29·14-s + 1.99·15-s − 0.491·16-s − 1.52·17-s − 12.6·18-s + 1.11·19-s + 2.12·20-s − 9.44·21-s − 7.49·22-s + 3.23·23-s − 7.44·24-s − 4.53·25-s + 2.26·26-s + 7.57·27-s − 10.0·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.69·3-s + 1.56·4-s + 0.304·5-s − 2.70·6-s − 1.21·7-s − 0.898·8-s + 1.86·9-s − 0.486·10-s + 0.998·11-s + 2.64·12-s − 0.277·13-s + 1.94·14-s + 0.514·15-s − 0.122·16-s − 0.369·17-s − 2.97·18-s + 0.255·19-s + 0.475·20-s − 2.05·21-s − 1.59·22-s + 0.673·23-s − 1.52·24-s − 0.907·25-s + 0.443·26-s + 1.45·27-s − 1.90·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 463 | \( 1 + T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 - 0.680T + 5T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 17 | \( 1 + 1.52T + 17T^{2} \) |
| 19 | \( 1 - 1.11T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 - 0.0198T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 - 3.84T + 43T^{2} \) |
| 47 | \( 1 + 9.40T + 47T^{2} \) |
| 53 | \( 1 + 6.20T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 - 5.81T + 83T^{2} \) |
| 89 | \( 1 + 5.39T + 89T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−7.893946635446765324856142083396, −7.22491630230203234176383394829, −6.82273872202189474978847614481, −6.01154796515955289400953539590, −4.60181429262375772368935865514, −3.51610544421459373595196060072, −3.11321345624189692988220587921, −2.03095630502001814283334099936, −1.51986004578180651311676032989, 0,
1.51986004578180651311676032989, 2.03095630502001814283334099936, 3.11321345624189692988220587921, 3.51610544421459373595196060072, 4.60181429262375772368935865514, 6.01154796515955289400953539590, 6.82273872202189474978847614481, 7.22491630230203234176383394829, 7.893946635446765324856142083396
