| L(s) = 1 | − 2.79·2-s + 1.08·3-s + 5.82·4-s − 3.02·6-s − 10.7·8-s − 1.82·9-s − 11-s + 6.31·12-s + 3.74·13-s + 18.3·16-s − 17-s + 5.11·18-s + 0.314·19-s + 2.79·22-s + 2.51·23-s − 11.5·24-s − 10.4·26-s − 5.22·27-s + 5.02·29-s − 8.42·31-s − 29.7·32-s − 1.08·33-s + 2.79·34-s − 10.6·36-s − 8.51·37-s − 0.880·38-s + 4.05·39-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 0.625·3-s + 2.91·4-s − 1.23·6-s − 3.78·8-s − 0.609·9-s − 0.301·11-s + 1.82·12-s + 1.03·13-s + 4.57·16-s − 0.242·17-s + 1.20·18-s + 0.0722·19-s + 0.596·22-s + 0.523·23-s − 2.36·24-s − 2.05·26-s − 1.00·27-s + 0.933·29-s − 1.51·31-s − 5.26·32-s − 0.188·33-s + 0.479·34-s − 1.77·36-s − 1.39·37-s − 0.142·38-s + 0.649·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 19 | \( 1 - 0.314T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 + 8.42T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 + 4.88T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 7.07T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 0.506T + 73T^{2} \) |
| 79 | \( 1 - 4.05T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 - 4.84T + 97T^{2} \) |
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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
−8.237889780990223898005607360773, −7.51936175483926290466393615753, −6.82621621518450016268125898777, −6.09988646038768649013293742442, −5.34518661884088843971282350444, −3.65885647577196103414715926419, −2.97389129430692063066758385677, −2.15981135627848165583811040111, −1.25150280510573613915901472739, 0,
1.25150280510573613915901472739, 2.15981135627848165583811040111, 2.97389129430692063066758385677, 3.65885647577196103414715926419, 5.34518661884088843971282350444, 6.09988646038768649013293742442, 6.82621621518450016268125898777, 7.51936175483926290466393615753, 8.237889780990223898005607360773
