L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.499i)12-s + (−0.499 − 0.866i)16-s + (−1.73 − i)23-s − 0.999i·27-s + (0.5 − 0.866i)31-s + 0.999i·33-s + 0.999·36-s + i·37-s − 0.999·44-s + (0.866 − 0.5i)47-s + 0.999i·48-s + (0.5 − 0.866i)49-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.499i)12-s + (−0.499 − 0.866i)16-s + (−1.73 − i)23-s − 0.999i·27-s + (0.5 − 0.866i)31-s + 0.999i·33-s + 0.999·36-s + i·37-s − 0.999·44-s + (0.866 − 0.5i)47-s + 0.999i·48-s + (0.5 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7499758754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7499758754\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
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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
−8.689533112582305679743833974213, −7.934409906643424644846018266745, −7.13396606402587707066788029906, −6.23226947785887894819255443348, −5.93228986506209758714463628409, −5.10602467989208173142266921819, −4.19964707562303371141548440401, −2.71049156770700380527076515308, −1.81241593553954901483905139104, −0.53115092429659283065575484937,
1.74284452052943568260306113996, 2.90835925064399848199713132426, 3.95710386230969074382491729595, 4.53069524800404915178777638018, 5.62088639152425492222226990586, 6.25186879635549421369336562786, 7.26415791938763102600590165139, 7.62895125750952989987674137709, 8.671557563247609180192398313553, 9.528327767847837080065683623844