| L(s) = 1 | − 2-s − 2.18·3-s + 4-s + 2.18·5-s + 2.18·6-s + 4.66·7-s − 8-s + 1.75·9-s − 2.18·10-s − 2.48·11-s − 2.18·12-s − 4.48·13-s − 4.66·14-s − 4.75·15-s + 16-s − 2.62·17-s − 1.75·18-s − 2.93·19-s + 2.18·20-s − 10.1·21-s + 2.48·22-s − 23-s + 2.18·24-s − 0.245·25-s + 4.48·26-s + 2.71·27-s + 4.66·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.25·3-s + 0.5·4-s + 0.975·5-s + 0.890·6-s + 1.76·7-s − 0.353·8-s + 0.584·9-s − 0.689·10-s − 0.749·11-s − 0.629·12-s − 1.24·13-s − 1.24·14-s − 1.22·15-s + 0.250·16-s − 0.637·17-s − 0.413·18-s − 0.673·19-s + 0.487·20-s − 2.22·21-s + 0.530·22-s − 0.208·23-s + 0.445·24-s − 0.0491·25-s + 0.879·26-s + 0.522·27-s + 0.881·28-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 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 - 2.18T + 5T^{2} \) |
| 7 | \( 1 - 4.66T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 + 1.33T + 61T^{2} \) |
| 67 | \( 1 - 0.628T + 67T^{2} \) |
| 71 | \( 1 - 1.50T + 71T^{2} \) |
| 73 | \( 1 - 3.63T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 - 7.86T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 8.72T + 97T^{2} \) |
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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
−9.301973440233441690106648517312, −8.414132626647474587063858082173, −7.56615140604597482113662362123, −6.81578103487490224136151927463, −5.64815293325255695566949608505, −5.32021733490648059013101135098, −4.44754608053833236539457620957, −2.36185942027396913971639140173, −1.66745421051693898853852100927, 0,
1.66745421051693898853852100927, 2.36185942027396913971639140173, 4.44754608053833236539457620957, 5.32021733490648059013101135098, 5.64815293325255695566949608505, 6.81578103487490224136151927463, 7.56615140604597482113662362123, 8.414132626647474587063858082173, 9.301973440233441690106648517312
