L(s) = 1 | + 1.73i·2-s + i·3-s − 1.99·4-s + (−0.866 − 0.5i)5-s − 1.73·6-s − i·7-s − 1.73i·8-s − 9-s + (0.866 − 1.49i)10-s − 11-s − 1.99i·12-s − i·13-s + 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯ |
L(s) = 1 | + 1.73i·2-s + i·3-s − 1.99·4-s + (−0.866 − 0.5i)5-s − 1.73·6-s − i·7-s − 1.73i·8-s − 9-s + (0.866 − 1.49i)10-s − 11-s − 1.99i·12-s − i·13-s + 1.73·14-s + (0.5 − 0.866i)15-s + 0.999·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06270380737\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06270380737\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.73iT - T^{2} \) |
| 13 | \( 1 + iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.73T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.716779984835265948294111284714, −8.759073303318826739932359933332, −8.036029158431686010515854727605, −7.73226514464078864908299341301, −6.59312752757852423803732211797, −5.65059145788940228130233170726, −4.76844259961484269743955896908, −4.32454306717423570288633879372, −3.25896897379239515047641494336, −0.05280534375367896970166015406,
1.99781546648243450477719369885, 2.47158519213021804123788461998, 3.53661093846183708486206301997, 4.55864405236426894536662985868, 5.74770087940335500721356019633, 6.81862783948448530626835031616, 7.83066427009607932193316082783, 8.661092456410526257751963938299, 9.188804961892295477213723660269, 10.46973282102024591446341778399