L(s) = 1 | + (0.258 + 0.965i)3-s + (0.866 + 0.5i)5-s + (0.965 + 0.258i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)15-s + i·21-s + (−0.965 + 1.67i)23-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)27-s − 1.73i·29-s + (0.707 + 0.707i)35-s − i·41-s + 0.517i·43-s − 45-s + (−0.707 + 1.22i)47-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)3-s + (0.866 + 0.5i)5-s + (0.965 + 0.258i)7-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)15-s + i·21-s + (−0.965 + 1.67i)23-s + (0.499 + 0.866i)25-s + (−0.707 − 0.707i)27-s − 1.73i·29-s + (0.707 + 0.707i)35-s − i·41-s + 0.517i·43-s − 45-s + (−0.707 + 1.22i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.709292343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.709292343\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.965 - 0.258i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + 1.73iT - T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + iT - T^{2} \) |
| 43 | \( 1 - 0.517iT - T^{2} \) |
| 47 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + 0.517T + T^{2} \) |
| 89 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.329365364357899190561372826612, −8.108236009971729565425277505568, −7.86753431482088266811542554214, −6.62979739172028667040735112761, −5.71650879980373000560719936272, −5.31852031421457396777609971016, −4.35537609784439611353900669960, −3.54365403883430324923092362188, −2.50329197888624356224929205153, −1.76319210846271758349500120481,
1.04057673472036721774141244882, 1.90862499772350490124523117414, 2.65517240350598773487149643906, 3.95285663889900818693210704024, 4.96614691673927239117209768428, 5.56807749772854513018227917887, 6.51133993663749231667371270871, 7.01985255672697251656384912516, 8.056868946771393468630288798186, 8.485160497189785135512760073430