| L(s) = 1 | + 2-s + 0.454·3-s + 4-s − 5-s + 0.454·6-s − 4.57·7-s + 8-s − 2.79·9-s − 10-s + 11-s + 0.454·12-s + 4.27·13-s − 4.57·14-s − 0.454·15-s + 16-s + 0.731·17-s − 2.79·18-s − 3.54·19-s − 20-s − 2.08·21-s + 22-s + 6.31·23-s + 0.454·24-s + 25-s + 4.27·26-s − 2.63·27-s − 4.57·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.262·3-s + 0.5·4-s − 0.447·5-s + 0.185·6-s − 1.73·7-s + 0.353·8-s − 0.931·9-s − 0.316·10-s + 0.301·11-s + 0.131·12-s + 1.18·13-s − 1.22·14-s − 0.117·15-s + 0.250·16-s + 0.177·17-s − 0.658·18-s − 0.812·19-s − 0.223·20-s − 0.454·21-s + 0.213·22-s + 1.31·23-s + 0.0928·24-s + 0.200·25-s + 0.837·26-s − 0.507·27-s − 0.865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.080461329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080461329\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 0.454T + 3T^{2} \) |
| 7 | \( 1 + 4.57T + 7T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 - 0.731T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 + 6.91T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 - 2.67T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 1.24T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 3.09T + 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−7.71012587140877415084168415667, −6.89813288699201749267435457223, −6.43419141423075475523938154691, −5.81453320956734270261454308550, −5.13117991211105543479623875089, −3.92257969931146114885599479696, −3.53185003617625185588419732923, −3.03560278949076655873658990812, −2.04522346341788248625213456811, −0.60364521797104617129996334226,
0.60364521797104617129996334226, 2.04522346341788248625213456811, 3.03560278949076655873658990812, 3.53185003617625185588419732923, 3.92257969931146114885599479696, 5.13117991211105543479623875089, 5.81453320956734270261454308550, 6.43419141423075475523938154691, 6.89813288699201749267435457223, 7.71012587140877415084168415667
