L(s) = 1 | − 2-s + 4-s − 3.85·5-s − 2.71·7-s − 8-s + 3.85·10-s + 0.310·11-s + 2.17·13-s + 2.71·14-s + 16-s + 0.173·17-s − 1.74·19-s − 3.85·20-s − 0.310·22-s − 0.551·23-s + 9.85·25-s − 2.17·26-s − 2.71·28-s + 1.97·29-s + 5.74·31-s − 32-s − 0.173·34-s + 10.4·35-s + 3.77·37-s + 1.74·38-s + 3.85·40-s − 0.377·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.72·5-s − 1.02·7-s − 0.353·8-s + 1.21·10-s + 0.0936·11-s + 0.602·13-s + 0.725·14-s + 0.250·16-s + 0.0421·17-s − 0.399·19-s − 0.861·20-s − 0.0662·22-s − 0.115·23-s + 1.97·25-s − 0.425·26-s − 0.513·28-s + 0.366·29-s + 1.03·31-s − 0.176·32-s − 0.0297·34-s + 1.76·35-s + 0.620·37-s + 0.282·38-s + 0.609·40-s − 0.0590·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
good | 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 - 0.310T + 11T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 17 | \( 1 - 0.173T + 17T^{2} \) |
| 19 | \( 1 + 1.74T + 19T^{2} \) |
| 23 | \( 1 + 0.551T + 23T^{2} \) |
| 29 | \( 1 - 1.97T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 - 3.77T + 37T^{2} \) |
| 41 | \( 1 + 0.377T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 2.59T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 4.88T + 97T^{2} \) |
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
−8.108066362221360855457564926934, −7.53922075534008444366876800523, −6.64629083156053741068168182050, −6.29001898753218656891850678338, −4.97925577212787589731440915062, −4.00188242574905114509137251438, −3.45367358356901074488987639488, −2.59130729687721254575211245552, −0.999145651603834286080813506855, 0,
0.999145651603834286080813506855, 2.59130729687721254575211245552, 3.45367358356901074488987639488, 4.00188242574905114509137251438, 4.97925577212787589731440915062, 6.29001898753218656891850678338, 6.64629083156053741068168182050, 7.53922075534008444366876800523, 8.108066362221360855457564926934