L(s) = 1 | + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + 2.44·6-s + (−0.898 − 0.898i)7-s + (2 − 2i)8-s − 2.99i·9-s − 13.7·11-s + (−2.44 − 2.44i)12-s + (−12.7 + 12.7i)13-s + 1.79i·14-s − 4·16-s + (−15.8 − 15.8i)17-s + (−2.99 + 2.99i)18-s + 25.7i·19-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + 0.408·6-s + (−0.128 − 0.128i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s − 1.25·11-s + (−0.204 − 0.204i)12-s + (−0.984 + 0.984i)13-s + 0.128i·14-s − 0.250·16-s + (−0.935 − 0.935i)17-s + (−0.166 + 0.166i)18-s + 1.35i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0483287 + 0.170124i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0483287 + 0.170124i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.898 + 0.898i)T + 49iT^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 + (12.7 - 12.7i)T - 169iT^{2} \) |
| 17 | \( 1 + (15.8 + 15.8i)T + 289iT^{2} \) |
| 19 | \( 1 - 25.7iT - 361T^{2} \) |
| 23 | \( 1 + (10.6 - 10.6i)T - 529iT^{2} \) |
| 29 | \( 1 + 25.7iT - 841T^{2} \) |
| 31 | \( 1 + 39.5T + 961T^{2} \) |
| 37 | \( 1 + (-27 - 27i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 17.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-12.4 + 12.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (9.30 + 9.30i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-19.0 + 19.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 20iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 15.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-48.0 - 48.0i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 6.20T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-37.2 + 37.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 115. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (82.2 - 82.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 117. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (81.9 + 81.9i)T + 9.40e3iT^{2} \) |
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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
−12.98110289934843233281428072140, −11.95047838264274956954805476474, −11.15444544549919937431452457772, −10.06893570799668144457871089459, −9.448853024049826765891812105246, −8.059914367064029189004579170825, −6.96493848515660313922851764773, −5.38035734775189934034884366767, −4.08437209567880076504784101286, −2.34636973686680268737516353970,
0.12932674921362904919394375637, 2.46916951884235372074449748693, 4.85287465499011537611036602765, 5.88835116358686602141676055990, 7.14342027454785715342519085476, 7.980092986599776365741395089227, 9.155079518718766339670725072935, 10.44301357574432506219193196522, 11.06210340680895822954254760922, 12.66041507798360231989396977534