| L(s) = 1 | + 1.24·2-s + 3-s − 0.445·4-s − 5-s + 1.24·6-s − 0.246·7-s − 3.04·8-s + 9-s − 1.24·10-s + 0.109·11-s − 0.445·12-s − 0.307·14-s − 15-s − 2.91·16-s − 1.50·17-s + 1.24·18-s + 3.44·19-s + 0.445·20-s − 0.246·21-s + 0.137·22-s − 7.80·23-s − 3.04·24-s + 25-s + 27-s + 0.109·28-s − 4.66·29-s − 1.24·30-s + ⋯ |
| L(s) = 1 | + 0.881·2-s + 0.577·3-s − 0.222·4-s − 0.447·5-s + 0.509·6-s − 0.0933·7-s − 1.07·8-s + 0.333·9-s − 0.394·10-s + 0.0331·11-s − 0.128·12-s − 0.0823·14-s − 0.258·15-s − 0.727·16-s − 0.365·17-s + 0.293·18-s + 0.790·19-s + 0.0995·20-s − 0.0538·21-s + 0.0292·22-s − 1.62·23-s − 0.622·24-s + 0.200·25-s + 0.192·27-s + 0.0207·28-s − 0.866·29-s − 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 7 | \( 1 + 0.246T + 7T^{2} \) |
| 11 | \( 1 - 0.109T + 11T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 + 7.80T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 - 0.939T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 0.978T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 - 7.92T + 83T^{2} \) |
| 89 | \( 1 - 1.62T + 89T^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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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
−8.260041205750820050298404835106, −8.031503479119975708216320310254, −6.83807996402292084023526942428, −6.15365676067489718953062717948, −5.14967863719663077994464267563, −4.48739965478170219925189236143, −3.61164013062978112014972415273, −3.09431259761080525220552955485, −1.81083073995611742153847986908, 0,
1.81083073995611742153847986908, 3.09431259761080525220552955485, 3.61164013062978112014972415273, 4.48739965478170219925189236143, 5.14967863719663077994464267563, 6.15365676067489718953062717948, 6.83807996402292084023526942428, 8.031503479119975708216320310254, 8.260041205750820050298404835106
