| L(s) = 1 | − 3-s + 3.23·5-s + 1.23·7-s + 9-s + 4·11-s − 4.47·13-s − 3.23·15-s − 7.23·17-s + 2.76·19-s − 1.23·21-s − 23-s + 5.47·25-s − 27-s + 4.47·29-s − 2.47·31-s − 4·33-s + 4.00·35-s + 4.47·37-s + 4.47·39-s + 6.94·41-s + 7.70·43-s + 3.23·45-s + 4·47-s − 5.47·49-s + 7.23·51-s + 0.763·53-s + 12.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.44·5-s + 0.467·7-s + 0.333·9-s + 1.20·11-s − 1.24·13-s − 0.835·15-s − 1.75·17-s + 0.634·19-s − 0.269·21-s − 0.208·23-s + 1.09·25-s − 0.192·27-s + 0.830·29-s − 0.444·31-s − 0.696·33-s + 0.676·35-s + 0.735·37-s + 0.716·39-s + 1.08·41-s + 1.17·43-s + 0.482·45-s + 0.583·47-s − 0.781·49-s + 1.01·51-s + 0.104·53-s + 1.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.282908287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.282908287\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 7.23T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.94T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 0.763T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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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
−8.507716418999817764574548290266, −7.40494721912235168355962595950, −6.75262703450261873663320916954, −6.18206608454870779948807654788, −5.45658927915618676565695952104, −4.72794266175950756363362897090, −4.06345242718058504815478985112, −2.55101066063760646660556144537, −1.98720633005334702452871899136, −0.898696741706639839845683108283,
0.898696741706639839845683108283, 1.98720633005334702452871899136, 2.55101066063760646660556144537, 4.06345242718058504815478985112, 4.72794266175950756363362897090, 5.45658927915618676565695952104, 6.18206608454870779948807654788, 6.75262703450261873663320916954, 7.40494721912235168355962595950, 8.507716418999817764574548290266
