L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.258 + 0.965i)3-s + (0.500 + 0.866i)6-s + (0.965 + 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.258 − 0.965i)13-s + (0.866 − 0.500i)14-s + 1.00·16-s + (0.258 + 0.965i)17-s + (−0.965 + 0.258i)18-s + (−0.866 + 0.5i)19-s + (−0.499 + 0.866i)21-s + (−0.866 + 0.5i)24-s + (−0.500 − 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.258 + 0.965i)3-s + (0.500 + 0.866i)6-s + (0.965 + 0.258i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (0.258 − 0.965i)13-s + (0.866 − 0.500i)14-s + 1.00·16-s + (0.258 + 0.965i)17-s + (−0.965 + 0.258i)18-s + (−0.866 + 0.5i)19-s + (−0.499 + 0.866i)21-s + (−0.866 + 0.5i)24-s + (−0.500 − 0.866i)26-s + (0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.623998999\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.623998999\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.965 - 0.258i)T \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−10.05470266199976570119778962133, −8.817778428943971704438896512118, −8.282220207184451499158823436510, −7.49449117897726055549467287743, −5.84224031396770577077822551973, −5.51546046867151206112909658997, −4.39832766630306848906170610681, −3.92881235075408855005909353873, −2.93678566040578532679537013896, −1.80503919206670076590381488672,
1.23447959103052203842511443275, 2.27961541601106450202869188209, 3.91134506199235148467388215251, 4.82764857528534299601012234913, 5.47182954764859678151774770133, 6.35211964297955901417087083761, 7.09037178307977937680674286625, 7.54721339914852121790445845387, 8.515909657754095051086779995012, 9.375931597112769792684853802081