L(s) = 1 | + 2-s − 0.293·3-s + 4-s + 2.91·5-s − 0.293·6-s − 1.20·7-s + 8-s − 2.91·9-s + 2.91·10-s + 1.41·11-s − 0.293·12-s − 4.65·13-s − 1.20·14-s − 0.854·15-s + 16-s − 2.36·17-s − 2.91·18-s − 6.18·19-s + 2.91·20-s + 0.352·21-s + 1.41·22-s − 23-s − 0.293·24-s + 3.49·25-s − 4.65·26-s + 1.73·27-s − 1.20·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.169·3-s + 0.5·4-s + 1.30·5-s − 0.119·6-s − 0.453·7-s + 0.353·8-s − 0.971·9-s + 0.921·10-s + 0.426·11-s − 0.0846·12-s − 1.29·13-s − 0.320·14-s − 0.220·15-s + 0.250·16-s − 0.574·17-s − 0.686·18-s − 1.41·19-s + 0.651·20-s + 0.0768·21-s + 0.301·22-s − 0.208·23-s − 0.0598·24-s + 0.699·25-s − 0.912·26-s + 0.333·27-s − 0.226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 0.293T + 3T^{2} \) |
| 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 + 6.18T + 19T^{2} \) |
| 29 | \( 1 - 7.99T + 29T^{2} \) |
| 31 | \( 1 - 0.618T + 31T^{2} \) |
| 37 | \( 1 + 8.40T + 37T^{2} \) |
| 41 | \( 1 - 9.57T + 41T^{2} \) |
| 43 | \( 1 - 5.03T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 + 8.36T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 + 3.54T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 + 0.327T + 79T^{2} \) |
| 83 | \( 1 + 3.85T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 9.13T + 97T^{2} \) |
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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
−7.55403104198987415414563138936, −6.60090623560683855508200670955, −6.26468202457819113764674626358, −5.69040390686996718079939709616, −4.83758027744672165629646341742, −4.27463378868899899802236163693, −2.96164218902943765853763194626, −2.52600914854840333135621384351, −1.65383199614932223583622041723, 0,
1.65383199614932223583622041723, 2.52600914854840333135621384351, 2.96164218902943765853763194626, 4.27463378868899899802236163693, 4.83758027744672165629646341742, 5.69040390686996718079939709616, 6.26468202457819113764674626358, 6.60090623560683855508200670955, 7.55403104198987415414563138936