L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.17 − 0.315i)5-s + (−0.707 + 0.707i)8-s − 9-s + 1.21·10-s + (−0.130 − 0.991i)13-s + (0.500 − 0.866i)16-s + (0.130 + 0.226i)17-s + (0.965 − 0.258i)18-s + (−1.17 + 0.315i)20-s + (0.417 + 0.241i)25-s + (0.382 + 0.923i)26-s + (0.965 + 1.67i)29-s + (−0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.17 − 0.315i)5-s + (−0.707 + 0.707i)8-s − 9-s + 1.21·10-s + (−0.130 − 0.991i)13-s + (0.500 − 0.866i)16-s + (0.130 + 0.226i)17-s + (0.965 − 0.258i)18-s + (−1.17 + 0.315i)20-s + (0.417 + 0.241i)25-s + (0.382 + 0.923i)26-s + (0.965 + 1.67i)29-s + (−0.258 + 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2409476770\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2409476770\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (0.130 + 0.991i)T \) |
good | 3 | \( 1 + T^{2} \) |
| 5 | \( 1 + (1.17 + 0.315i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 37 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (1.91 + 0.513i)T + (0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 - 1.58iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (1.53 - 0.410i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.478 - 1.78i)T + (-0.866 + 0.5i)T^{2} \) |
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
−8.965018596184294875409134488718, −8.547267015924319275119492440249, −7.968229026462835981901761098588, −7.31522655822889284942978710336, −6.43084377861565523545734855768, −5.54037040525370852952476693406, −4.78837143492362589180903728589, −3.42775522796061511811109942993, −2.77167420031668173756156906995, −1.18771722730415636478809701578,
0.23659286314471220515139068177, 1.97183649968140835903726258331, 3.00200025523759241832328764188, 3.73255511884505025519744355038, 4.73411889114349590040707239737, 6.05343035827264798286357389692, 6.73392508708023373119945620798, 7.53000420155429102213624010058, 8.174867591214984942794519681052, 8.689214384040009448060020346077