L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−1 + 1.73i)7-s − 0.999·8-s + (1 + 1.73i)9-s + 0.999·10-s + (−0.499 − 0.866i)12-s + (−2.5 + 4.33i)13-s − 1.99·14-s + (0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)18-s + (−1 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s − 0.408·6-s + (−0.377 + 0.654i)7-s − 0.353·8-s + (0.333 + 0.577i)9-s + 0.316·10-s + (−0.144 − 0.249i)12-s + (−0.693 + 1.20i)13-s − 0.534·14-s + (0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.235 + 0.408i)18-s + (−0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.469758 + 1.18819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.469758 + 1.18819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-5.5 + 2.59i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−11.96463674589906864746363785777, −10.81035214955022354325011430647, −9.717101988542977364924752327773, −9.076344566212397475217116691744, −7.965237867784137235117120089834, −6.86304654964215497025734217045, −5.84544051670146172766254244245, −4.92055442618148019333157482523, −4.07376199259696496572100081909, −2.30869286353212676934636417154,
0.818423107230496547229910307170, 2.61686854403444419943394780119, 3.77203135512881464042904972953, 5.08581985468857083352068928180, 6.29812262467377556250580749964, 7.02975292160033281840871521809, 8.172755845669916715448094422866, 9.688367465214071462326753395837, 10.12086597330884111849399998088, 11.13664159154868711867188999518