| L(s) = 1 | + 2.41·2-s − 3-s + 3.82·4-s − 5-s − 2.41·6-s + 3.41·7-s + 4.41·8-s + 9-s − 2.41·10-s − 1.41·11-s − 3.82·12-s + 2.58·13-s + 8.24·14-s + 15-s + 2.99·16-s − 6.82·17-s + 2.41·18-s + 19-s − 3.82·20-s − 3.41·21-s − 3.41·22-s − 3.65·23-s − 4.41·24-s + 25-s + 6.24·26-s − 27-s + 13.0·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.447·5-s − 0.985·6-s + 1.29·7-s + 1.56·8-s + 0.333·9-s − 0.763·10-s − 0.426·11-s − 1.10·12-s + 0.717·13-s + 2.20·14-s + 0.258·15-s + 0.749·16-s − 1.65·17-s + 0.569·18-s + 0.229·19-s − 0.856·20-s − 0.745·21-s − 0.727·22-s − 0.762·23-s − 0.901·24-s + 0.200·25-s + 1.22·26-s − 0.192·27-s + 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.712813034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.712813034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 5.07T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 6.48T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−11.94422651748311600022830476308, −11.10329910737413387041482926454, −10.76851606133144051168658296443, −8.780432841597661212279461053054, −7.59207325691769332581278216658, −6.56302654898553469937207572746, −5.48498184650769896560590443491, −4.67595605683700315639815054953, −3.82041776524910840198178033245, −2.08829373944119483464532167414,
2.08829373944119483464532167414, 3.82041776524910840198178033245, 4.67595605683700315639815054953, 5.48498184650769896560590443491, 6.56302654898553469937207572746, 7.59207325691769332581278216658, 8.780432841597661212279461053054, 10.76851606133144051168658296443, 11.10329910737413387041482926454, 11.94422651748311600022830476308
