L(s) = 1 | + 1.48·2-s + 0.207·4-s − 2.74·5-s + 0.513·7-s − 2.66·8-s − 4.07·10-s − 4.11·11-s + 3.97·13-s + 0.763·14-s − 4.37·16-s − 0.363·17-s − 19-s − 0.568·20-s − 6.10·22-s + 2.05·23-s + 2.53·25-s + 5.90·26-s + 0.106·28-s − 10.2·29-s − 1.54·31-s − 1.16·32-s − 0.540·34-s − 1.41·35-s − 8.51·37-s − 1.48·38-s + 7.31·40-s + 0.554·41-s + ⋯ |
L(s) = 1 | + 1.05·2-s + 0.103·4-s − 1.22·5-s + 0.194·7-s − 0.941·8-s − 1.28·10-s − 1.23·11-s + 1.10·13-s + 0.203·14-s − 1.09·16-s − 0.0882·17-s − 0.229·19-s − 0.127·20-s − 1.30·22-s + 0.428·23-s + 0.507·25-s + 1.15·26-s + 0.0201·28-s − 1.91·29-s − 0.277·31-s − 0.206·32-s − 0.0926·34-s − 0.238·35-s − 1.40·37-s − 0.241·38-s + 1.15·40-s + 0.0865·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.391487381\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.391487381\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 - 0.513T + 7T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 - 3.97T + 13T^{2} \) |
| 17 | \( 1 + 0.363T + 17T^{2} \) |
| 23 | \( 1 - 2.05T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 0.554T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 53 | \( 1 - 8.88T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 3.66T + 67T^{2} \) |
| 71 | \( 1 + 2.71T + 71T^{2} \) |
| 73 | \( 1 - 3.53T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 8.75T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63833010989812804312861209992, −7.27527598754769709174176494567, −6.20619071663477384789718565508, −5.58126612473796628211812869749, −4.97764128659095100635295352721, −4.19355262013241211246914078078, −3.66229978571046287074213410138, −3.08290846435169321162002137036, −2.01489053458907265163868434072, −0.47663984649143606775890001074,
0.47663984649143606775890001074, 2.01489053458907265163868434072, 3.08290846435169321162002137036, 3.66229978571046287074213410138, 4.19355262013241211246914078078, 4.97764128659095100635295352721, 5.58126612473796628211812869749, 6.20619071663477384789718565508, 7.27527598754769709174176494567, 7.63833010989812804312861209992