L(s) = 1 | − 1.15·2-s − 0.659·4-s + 0.421·5-s − 0.645·7-s + 3.07·8-s − 0.487·10-s + 0.640·11-s − 4.07·13-s + 0.747·14-s − 2.24·16-s − 6.47·17-s − 4.15·19-s − 0.277·20-s − 0.741·22-s + 7.48·23-s − 4.82·25-s + 4.71·26-s + 0.425·28-s + 0.298·29-s + 10.2·31-s − 3.55·32-s + 7.49·34-s − 0.271·35-s − 6.30·37-s + 4.80·38-s + 1.29·40-s − 3.79·41-s + ⋯ |
L(s) = 1 | − 0.818·2-s − 0.329·4-s + 0.188·5-s − 0.243·7-s + 1.08·8-s − 0.154·10-s + 0.192·11-s − 1.12·13-s + 0.199·14-s − 0.561·16-s − 1.57·17-s − 0.952·19-s − 0.0621·20-s − 0.158·22-s + 1.56·23-s − 0.964·25-s + 0.924·26-s + 0.0803·28-s + 0.0553·29-s + 1.83·31-s − 0.628·32-s + 1.28·34-s − 0.0459·35-s − 1.03·37-s + 0.779·38-s + 0.205·40-s − 0.592·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6443991559\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6443991559\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 5 | \( 1 - 0.421T + 5T^{2} \) |
| 7 | \( 1 + 0.645T + 7T^{2} \) |
| 11 | \( 1 - 0.640T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 0.298T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.30T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 - 0.987T + 43T^{2} \) |
| 47 | \( 1 + 0.803T + 47T^{2} \) |
| 53 | \( 1 - 0.867T + 53T^{2} \) |
| 59 | \( 1 + 0.841T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 - 3.61T + 67T^{2} \) |
| 71 | \( 1 - 4.80T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 - 9.06T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−8.336874719831389878971152926403, −7.73684220822253840949347381265, −6.76279752271544558541289206991, −6.47524079422901536295595041670, −5.06402281396394348227254495710, −4.72400628711162006757687814568, −3.81574510819215023068378756180, −2.61399124107256459014430297326, −1.80493294679015455105316572231, −0.49149542253428544341679125791,
0.49149542253428544341679125791, 1.80493294679015455105316572231, 2.61399124107256459014430297326, 3.81574510819215023068378756180, 4.72400628711162006757687814568, 5.06402281396394348227254495710, 6.47524079422901536295595041670, 6.76279752271544558541289206991, 7.73684220822253840949347381265, 8.336874719831389878971152926403