| L(s) = 1 | + 2.38·2-s − 3-s + 3.69·4-s − 1.70·5-s − 2.38·6-s + 3.54·7-s + 4.03·8-s + 9-s − 4.06·10-s + 1.77·11-s − 3.69·12-s − 3.57·13-s + 8.46·14-s + 1.70·15-s + 2.23·16-s − 17-s + 2.38·18-s − 2.97·19-s − 6.29·20-s − 3.54·21-s + 4.22·22-s − 9.54·23-s − 4.03·24-s − 2.09·25-s − 8.51·26-s − 27-s + 13.0·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s − 0.577·3-s + 1.84·4-s − 0.762·5-s − 0.973·6-s + 1.34·7-s + 1.42·8-s + 0.333·9-s − 1.28·10-s + 0.534·11-s − 1.06·12-s − 0.990·13-s + 2.26·14-s + 0.440·15-s + 0.559·16-s − 0.242·17-s + 0.562·18-s − 0.682·19-s − 1.40·20-s − 0.774·21-s + 0.901·22-s − 1.98·23-s − 0.823·24-s − 0.418·25-s − 1.67·26-s − 0.192·27-s + 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 5 | \( 1 + 1.70T + 5T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 3.57T + 13T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 + 9.54T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 3.61T + 37T^{2} \) |
| 41 | \( 1 + 1.21T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 + 5.08T + 47T^{2} \) |
| 53 | \( 1 - 3.39T + 53T^{2} \) |
| 59 | \( 1 + 1.11T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 + 7.67T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 - 0.700T + 79T^{2} \) |
| 83 | \( 1 - 8.64T + 83T^{2} \) |
| 89 | \( 1 - 8.82T + 89T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−7.29754332698160097305148852702, −6.58314520510186685782749077051, −5.90346832482534560990263877515, −5.22932519896108652024538682831, −4.51477324958363420036867224725, −4.24319370281517188938788524024, −3.49592866243890353851462631709, −2.28589459291503461483901830659, −1.69912838452194452806315818279, 0,
1.69912838452194452806315818279, 2.28589459291503461483901830659, 3.49592866243890353851462631709, 4.24319370281517188938788524024, 4.51477324958363420036867224725, 5.22932519896108652024538682831, 5.90346832482534560990263877515, 6.58314520510186685782749077051, 7.29754332698160097305148852702
