| L(s) = 1 | + 1.86·2-s + 3-s + 1.48·4-s − 4.30·5-s + 1.86·6-s − 7-s − 0.956·8-s + 9-s − 8.04·10-s − 3.88·11-s + 1.48·12-s − 1.86·14-s − 4.30·15-s − 4.76·16-s + 6.82·17-s + 1.86·18-s + 5.88·19-s − 6.41·20-s − 21-s − 7.26·22-s − 1.28·23-s − 0.956·24-s + 13.5·25-s + 27-s − 1.48·28-s + 1.83·29-s − 8.04·30-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.577·3-s + 0.743·4-s − 1.92·5-s + 0.762·6-s − 0.377·7-s − 0.338·8-s + 0.333·9-s − 2.54·10-s − 1.17·11-s + 0.429·12-s − 0.499·14-s − 1.11·15-s − 1.19·16-s + 1.65·17-s + 0.440·18-s + 1.35·19-s − 1.43·20-s − 0.218·21-s − 1.54·22-s − 0.267·23-s − 0.195·24-s + 2.71·25-s + 0.192·27-s − 0.281·28-s + 0.339·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.626489542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626489542\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.86T + 2T^{2} \) |
| 5 | \( 1 + 4.30T + 5T^{2} \) |
| 11 | \( 1 + 3.88T + 11T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 - 1.37T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 6.71T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 + 9.83T + 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 - 3.34T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 - 4.78T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 - 3.87T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−8.230954600862016270214301632398, −7.66221997676310517005374754211, −7.34657797952186842308527081548, −6.14203960628414145496159815831, −5.31469312675550016321381527147, −4.55125215101759367944588369813, −3.91070602171223859274677745305, −3.05956609331685351365113178670, −2.88809540236873925491686937877, −0.74698543513622965215428588044,
0.74698543513622965215428588044, 2.88809540236873925491686937877, 3.05956609331685351365113178670, 3.91070602171223859274677745305, 4.55125215101759367944588369813, 5.31469312675550016321381527147, 6.14203960628414145496159815831, 7.34657797952186842308527081548, 7.66221997676310517005374754211, 8.230954600862016270214301632398
