| L(s) = 1 | − 3-s + 1.59·5-s + 3.02·7-s + 9-s + 2.27·11-s + 6.72·13-s − 1.59·15-s + 0.393·17-s − 3.65·19-s − 3.02·21-s − 23-s − 2.46·25-s − 27-s − 29-s + 7.95·31-s − 2.27·33-s + 4.81·35-s − 4.01·37-s − 6.72·39-s + 5.18·41-s + 3.72·43-s + 1.59·45-s + 1.48·47-s + 2.15·49-s − 0.393·51-s + 6.75·53-s + 3.62·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.712·5-s + 1.14·7-s + 0.333·9-s + 0.685·11-s + 1.86·13-s − 0.411·15-s + 0.0953·17-s − 0.837·19-s − 0.660·21-s − 0.208·23-s − 0.492·25-s − 0.192·27-s − 0.185·29-s + 1.42·31-s − 0.395·33-s + 0.814·35-s − 0.660·37-s − 1.07·39-s + 0.809·41-s + 0.568·43-s + 0.237·45-s + 0.217·47-s + 0.307·49-s − 0.0550·51-s + 0.927·53-s + 0.488·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814141315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814141315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 - 0.393T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 31 | \( 1 - 7.95T + 31T^{2} \) |
| 37 | \( 1 + 4.01T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 - 6.75T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 8.85T + 61T^{2} \) |
| 67 | \( 1 + 2.09T + 67T^{2} \) |
| 71 | \( 1 + 0.475T + 71T^{2} \) |
| 73 | \( 1 + 4.12T + 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 - 9.75T + 89T^{2} \) |
| 97 | \( 1 - 5.15T + 97T^{2} \) |
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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
−7.964910192233848132016856893421, −6.96632458020892464867473272862, −6.30116183515027647525542859274, −5.83193753160328536199035693751, −5.17835683701166735038140479147, −4.20591362761888587104069036813, −3.83184838376494127395589993099, −2.44379915172299274898563087651, −1.59181841664133979803120016049, −0.957822411530910297722503837395,
0.957822411530910297722503837395, 1.59181841664133979803120016049, 2.44379915172299274898563087651, 3.83184838376494127395589993099, 4.20591362761888587104069036813, 5.17835683701166735038140479147, 5.83193753160328536199035693751, 6.30116183515027647525542859274, 6.96632458020892464867473272862, 7.964910192233848132016856893421
