| L(s) = 1 | + 1.41·3-s − 1.48·5-s − 0.732·7-s − 0.999·9-s + 0.732·11-s + 2.86·13-s − 2.09·15-s + 1.55·17-s + 0.0963·19-s − 1.03·21-s − 2.80·25-s − 5.65·27-s − 6.96·29-s − 0.428·31-s + 1.03·33-s + 1.08·35-s + 7.81·37-s + 4.04·39-s + 2.46·41-s + 11.8·43-s + 1.48·45-s − 11.6·47-s − 6.46·49-s + 2.19·51-s + 10.0·53-s − 1.08·55-s + 0.136·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.662·5-s − 0.276·7-s − 0.333·9-s + 0.220·11-s + 0.794·13-s − 0.541·15-s + 0.376·17-s + 0.0221·19-s − 0.225·21-s − 0.560·25-s − 1.08·27-s − 1.29·29-s − 0.0770·31-s + 0.180·33-s + 0.183·35-s + 1.28·37-s + 0.648·39-s + 0.384·41-s + 1.80·43-s + 0.220·45-s − 1.69·47-s − 0.923·49-s + 0.307·51-s + 1.37·53-s − 0.146·55-s + 0.0180·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 0.0963T + 19T^{2} \) |
| 29 | \( 1 + 6.96T + 29T^{2} \) |
| 31 | \( 1 + 0.428T + 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 0.571T + 59T^{2} \) |
| 61 | \( 1 - 4.36T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 - 0.615T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 + 8.43T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−7.67034316452453692534272056572, −6.89927527335498121364709746717, −5.97346481766996957647476088510, −5.54739625337777023522828719475, −4.30029401118911711212395110552, −3.81846500166725876021972717806, −3.15196803262690178348692400100, −2.37985082259238084213132853844, −1.30695076192846539133152051762, 0,
1.30695076192846539133152051762, 2.37985082259238084213132853844, 3.15196803262690178348692400100, 3.81846500166725876021972717806, 4.30029401118911711212395110552, 5.54739625337777023522828719475, 5.97346481766996957647476088510, 6.89927527335498121364709746717, 7.67034316452453692534272056572
