L(s) = 1 | + 3.23·5-s − 7-s + 6.47·11-s − 0.763·13-s − 4.47·17-s − 1.23·19-s + 4·23-s + 5.47·25-s − 4.47·29-s + 2.47·31-s − 3.23·35-s + 4.47·37-s + 8.47·41-s + 6.47·43-s + 10.4·47-s + 49-s − 10·53-s + 20.9·55-s + 9.23·59-s − 11.2·61-s − 2.47·65-s + 4·67-s − 4.94·71-s − 2.94·73-s − 6.47·77-s − 12.9·79-s + 9.23·83-s + ⋯ |
L(s) = 1 | + 1.44·5-s − 0.377·7-s + 1.95·11-s − 0.211·13-s − 1.08·17-s − 0.283·19-s + 0.834·23-s + 1.09·25-s − 0.830·29-s + 0.444·31-s − 0.546·35-s + 0.735·37-s + 1.32·41-s + 0.986·43-s + 1.52·47-s + 0.142·49-s − 1.37·53-s + 2.82·55-s + 1.20·59-s − 1.43·61-s − 0.306·65-s + 0.488·67-s − 0.586·71-s − 0.344·73-s − 0.737·77-s − 1.45·79-s + 1.01·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.783930735\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.783930735\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 9.23T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.920868537315266251190266548871, −7.55697032260175243964962435700, −6.75194237814926577108489139463, −6.23801249531296123197739089397, −5.73137956258066710161386631477, −4.60349033228626955811018317994, −3.94941673211153744495821271853, −2.77162877342586045691452326091, −1.96462480003166355943015198923, −1.01446912297689590772616054098,
1.01446912297689590772616054098, 1.96462480003166355943015198923, 2.77162877342586045691452326091, 3.94941673211153744495821271853, 4.60349033228626955811018317994, 5.73137956258066710161386631477, 6.23801249531296123197739089397, 6.75194237814926577108489139463, 7.55697032260175243964962435700, 8.920868537315266251190266548871