| L(s) = 1 | − 2-s + 3-s + 4-s + 3.17·5-s − 6-s + 0.381·7-s − 8-s + 9-s − 3.17·10-s + 2.66·11-s + 12-s + 7.04·13-s − 0.381·14-s + 3.17·15-s + 16-s + 4.68·17-s − 18-s + 3.17·20-s + 0.381·21-s − 2.66·22-s + 1.43·23-s − 24-s + 5.08·25-s − 7.04·26-s + 27-s + 0.381·28-s − 8.20·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.42·5-s − 0.408·6-s + 0.144·7-s − 0.353·8-s + 0.333·9-s − 1.00·10-s + 0.803·11-s + 0.288·12-s + 1.95·13-s − 0.102·14-s + 0.819·15-s + 0.250·16-s + 1.13·17-s − 0.235·18-s + 0.710·20-s + 0.0833·21-s − 0.568·22-s + 0.298·23-s − 0.204·24-s + 1.01·25-s − 1.38·26-s + 0.192·27-s + 0.0721·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.506279488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.506279488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.17T + 5T^{2} \) |
| 7 | \( 1 - 0.381T + 7T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 - 7.04T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 + 8.20T + 29T^{2} \) |
| 31 | \( 1 + 8.62T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 3.25T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 - 5.35T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 5.27T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 3.84T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 + 4.31T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−9.142212435830376800418041065992, −8.543432069546720679909830110048, −7.65940718702190718306186139679, −6.75416856517475322806471290136, −5.97963984231428766644631961072, −5.44169581815021083493183326157, −3.88605274228050051865060138207, −3.14811378314483333207779439206, −1.76708147163391172713260064548, −1.36857572622759936521182748289,
1.36857572622759936521182748289, 1.76708147163391172713260064548, 3.14811378314483333207779439206, 3.88605274228050051865060138207, 5.44169581815021083493183326157, 5.97963984231428766644631961072, 6.75416856517475322806471290136, 7.65940718702190718306186139679, 8.543432069546720679909830110048, 9.142212435830376800418041065992
