| L(s) = 1 | − 0.677·2-s + 2.35·3-s − 1.54·4-s + 5-s − 1.59·6-s + 4.27·7-s + 2.39·8-s + 2.55·9-s − 0.677·10-s + 1.65·11-s − 3.63·12-s + 6.21·13-s − 2.89·14-s + 2.35·15-s + 1.45·16-s − 1.73·18-s − 3.38·19-s − 1.54·20-s + 10.0·21-s − 1.12·22-s − 4.67·23-s + 5.65·24-s + 25-s − 4.21·26-s − 1.04·27-s − 6.59·28-s − 3.64·29-s + ⋯ |
| L(s) = 1 | − 0.479·2-s + 1.36·3-s − 0.770·4-s + 0.447·5-s − 0.652·6-s + 1.61·7-s + 0.848·8-s + 0.852·9-s − 0.214·10-s + 0.499·11-s − 1.04·12-s + 1.72·13-s − 0.774·14-s + 0.608·15-s + 0.363·16-s − 0.408·18-s − 0.777·19-s − 0.344·20-s + 2.20·21-s − 0.239·22-s − 0.974·23-s + 1.15·24-s + 0.200·25-s − 0.825·26-s − 0.200·27-s − 1.24·28-s − 0.677·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.453310853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453310853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.677T + 2T^{2} \) |
| 3 | \( 1 - 2.35T + 3T^{2} \) |
| 7 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 6.21T + 13T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 - 2.99T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 43 | \( 1 + 0.998T + 43T^{2} \) |
| 47 | \( 1 - 2.00T + 47T^{2} \) |
| 53 | \( 1 + 6.95T + 53T^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 - 5.80T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 9.92T + 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 2.69T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.162796694739142168722185139209, −8.665369377535030598278340469009, −8.229622705654056691411158998504, −7.64335403338169262578422007882, −6.30967281965633643943301479420, −5.23039966123757210138317378198, −4.21175681618110629774846919109, −3.60697318578940013078561454011, −2.02785945716917220909078254875, −1.37215013044830693650744149421,
1.37215013044830693650744149421, 2.02785945716917220909078254875, 3.60697318578940013078561454011, 4.21175681618110629774846919109, 5.23039966123757210138317378198, 6.30967281965633643943301479420, 7.64335403338169262578422007882, 8.229622705654056691411158998504, 8.665369377535030598278340469009, 9.162796694739142168722185139209
