| L(s) = 1 | − 1.86·2-s − 3-s + 1.46·4-s + 1.32·5-s + 1.86·6-s + 8-s + 9-s − 2.46·10-s + 11-s − 1.46·12-s + 0.398·13-s − 1.32·15-s − 4.78·16-s − 6.64·17-s − 1.86·18-s + 1.32·19-s + 1.93·20-s − 1.86·22-s − 24-s − 3.24·25-s − 0.740·26-s − 27-s + 1.47·29-s + 2.46·30-s − 4.79·31-s + 6.90·32-s − 33-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.731·4-s + 0.591·5-s + 0.759·6-s + 0.353·8-s + 0.333·9-s − 0.778·10-s + 0.301·11-s − 0.422·12-s + 0.110·13-s − 0.341·15-s − 1.19·16-s − 1.61·17-s − 0.438·18-s + 0.303·19-s + 0.432·20-s − 0.396·22-s − 0.204·24-s − 0.649·25-s − 0.145·26-s − 0.192·27-s + 0.273·29-s + 0.449·30-s − 0.861·31-s + 1.22·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 13 | \( 1 - 0.398T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 - 1.32T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 + 1.60T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 4.79T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 - 1.45T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 + 0.796T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−9.171202391885787833116613381744, −8.398336613963437344716494271755, −7.45229637028583196808411687795, −6.74059158054450055615956028614, −5.97422091086996797554416561203, −4.92653300500402358461728872591, −4.00284528451595776133018720756, −2.34460998863420988369674002544, −1.40526693138977516425719040294, 0,
1.40526693138977516425719040294, 2.34460998863420988369674002544, 4.00284528451595776133018720756, 4.92653300500402358461728872591, 5.97422091086996797554416561203, 6.74059158054450055615956028614, 7.45229637028583196808411687795, 8.398336613963437344716494271755, 9.171202391885787833116613381744
