| L(s) = 1 | − 2.62·2-s − 1.54·3-s + 4.87·4-s + 5-s + 4.04·6-s + 0.0125·7-s − 7.54·8-s − 0.625·9-s − 2.62·10-s + 0.372·11-s − 7.51·12-s + 3.24·13-s − 0.0329·14-s − 1.54·15-s + 10.0·16-s + 17-s + 1.64·18-s − 7.23·19-s + 4.87·20-s − 0.0193·21-s − 0.977·22-s + 2.15·23-s + 11.6·24-s + 25-s − 8.50·26-s + 5.58·27-s + 0.0613·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.889·3-s + 2.43·4-s + 0.447·5-s + 1.64·6-s + 0.00475·7-s − 2.66·8-s − 0.208·9-s − 0.829·10-s + 0.112·11-s − 2.16·12-s + 0.899·13-s − 0.00881·14-s − 0.397·15-s + 2.50·16-s + 0.242·17-s + 0.386·18-s − 1.65·19-s + 1.09·20-s − 0.00423·21-s − 0.208·22-s + 0.448·23-s + 2.37·24-s + 0.200·25-s − 1.66·26-s + 1.07·27-s + 0.0115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 1.54T + 3T^{2} \) |
| 7 | \( 1 - 0.0125T + 7T^{2} \) |
| 11 | \( 1 - 0.372T + 11T^{2} \) |
| 13 | \( 1 - 3.24T + 13T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 + 0.852T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 + 1.45T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 5.85T + 83T^{2} \) |
| 89 | \( 1 + 9.05T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−7.995959377572758312705519504314, −6.92782033855336739621405991187, −6.53311076503401441631945466598, −5.91933780394446114083358073886, −5.21451139633391239882246065940, −3.92600925880690653712224174760, −2.77828353246944548594743572554, −1.90867278212828234134782357884, −1.01388897813462347326604194634, 0,
1.01388897813462347326604194634, 1.90867278212828234134782357884, 2.77828353246944548594743572554, 3.92600925880690653712224174760, 5.21451139633391239882246065940, 5.91933780394446114083358073886, 6.53311076503401441631945466598, 6.92782033855336739621405991187, 7.995959377572758312705519504314
