L(s) = 1 | + 3.00·3-s − 1.92·5-s + 7-s + 6.02·9-s + 11-s + 13-s − 5.77·15-s − 1.45·17-s + 2.52·19-s + 3.00·21-s + 1.34·23-s − 1.30·25-s + 9.09·27-s + 8.20·29-s − 2.65·31-s + 3.00·33-s − 1.92·35-s + 6.81·37-s + 3.00·39-s − 8.51·41-s + 0.841·43-s − 11.5·45-s + 10.9·47-s + 49-s − 4.36·51-s + 4.76·53-s − 1.92·55-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.859·5-s + 0.377·7-s + 2.00·9-s + 0.301·11-s + 0.277·13-s − 1.49·15-s − 0.352·17-s + 0.578·19-s + 0.655·21-s + 0.281·23-s − 0.260·25-s + 1.75·27-s + 1.52·29-s − 0.476·31-s + 0.523·33-s − 0.324·35-s + 1.12·37-s + 0.481·39-s − 1.33·41-s + 0.128·43-s − 1.72·45-s + 1.60·47-s + 0.142·49-s − 0.611·51-s + 0.654·53-s − 0.259·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.981402465\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.981402465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 - 2.52T + 19T^{2} \) |
| 23 | \( 1 - 1.34T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 + 2.65T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 + 8.51T + 41T^{2} \) |
| 43 | \( 1 - 0.841T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.76T + 53T^{2} \) |
| 59 | \( 1 - 0.793T + 59T^{2} \) |
| 61 | \( 1 - 0.0741T + 61T^{2} \) |
| 67 | \( 1 + 4.41T + 67T^{2} \) |
| 71 | \( 1 + 3.15T + 71T^{2} \) |
| 73 | \( 1 - 8.22T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 0.0518T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 4.86T + 97T^{2} \) |
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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
−7.918214663577149982690507265209, −7.36555117273138592469051658739, −6.82148587266293620997684733562, −5.76020993162858455667175984991, −4.63381051283025263281755374311, −4.15208352381047131995004612005, −3.44030751900459455032240061775, −2.80397472555622583606931512413, −1.94920417696504611140157108095, −0.944046027389977591438343453492,
0.944046027389977591438343453492, 1.94920417696504611140157108095, 2.80397472555622583606931512413, 3.44030751900459455032240061775, 4.15208352381047131995004612005, 4.63381051283025263281755374311, 5.76020993162858455667175984991, 6.82148587266293620997684733562, 7.36555117273138592469051658739, 7.918214663577149982690507265209