| L(s) = 1 | − 2-s − 2.54·3-s + 4-s − 0.504·5-s + 2.54·6-s + 2.12·7-s − 8-s + 3.47·9-s + 0.504·10-s − 0.850·11-s − 2.54·12-s − 5.05·13-s − 2.12·14-s + 1.28·15-s + 16-s + 6.51·17-s − 3.47·18-s − 4.69·19-s − 0.504·20-s − 5.39·21-s + 0.850·22-s + 23-s + 2.54·24-s − 4.74·25-s + 5.05·26-s − 1.21·27-s + 2.12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.225·5-s + 1.03·6-s + 0.801·7-s − 0.353·8-s + 1.15·9-s + 0.159·10-s − 0.256·11-s − 0.734·12-s − 1.40·13-s − 0.566·14-s + 0.331·15-s + 0.250·16-s + 1.57·17-s − 0.819·18-s − 1.07·19-s − 0.112·20-s − 1.17·21-s + 0.181·22-s + 0.208·23-s + 0.519·24-s − 0.949·25-s + 0.990·26-s − 0.232·27-s + 0.400·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 0.504T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 + 0.850T + 11T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.58T + 37T^{2} \) |
| 41 | \( 1 + 8.56T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 + 0.904T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 1.16T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 - 7.30T + 79T^{2} \) |
| 83 | \( 1 - 6.12T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−9.442399869350856338822497632735, −8.127183348717062034388412655402, −7.73101552780892887086896463689, −6.74243999836949816396682926144, −5.91847855967045210766437037775, −5.09942579779263774265585368354, −4.37902096165342156751143929137, −2.70644420108074226899315811588, −1.30921993052179508683228468690, 0,
1.30921993052179508683228468690, 2.70644420108074226899315811588, 4.37902096165342156751143929137, 5.09942579779263774265585368354, 5.91847855967045210766437037775, 6.74243999836949816396682926144, 7.73101552780892887086896463689, 8.127183348717062034388412655402, 9.442399869350856338822497632735
