L(s) = 1 | − i·2-s − i·3-s − 4-s + (−1 − 2i)5-s − 6-s + i·8-s − 9-s + (−2 + i)10-s + 2·11-s + i·12-s − 6i·13-s + (−2 + i)15-s + 16-s + 4i·17-s + i·18-s − 6·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s + (−0.632 + 0.316i)10-s + 0.603·11-s + 0.288i·12-s − 1.66i·13-s + (−0.516 + 0.258i)15-s + 0.250·16-s + 0.970i·17-s + 0.235i·18-s − 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6729095314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6729095314\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.761751162750568859363025610708, −8.345444419641659704237362257480, −7.64889363709902841040397686416, −6.38505691649570084556754924588, −5.62634420081571443108872283508, −4.54127714516856148943948889454, −3.79205522420904522971850621723, −2.60102075713141769202822453220, −1.40952101175636141981813424684, −0.27455764657450222375701516397,
2.03881279619832219134759600815, 3.50732513391559170873267986052, 4.10231931860121892725461323508, 5.02980922003753405234051350815, 6.16296526541492960489266925645, 6.83557115096762224016837734688, 7.43150341820798187515509074942, 8.496411945930032903184952838663, 9.243029968623922934561804248419, 9.817672393704331028914981915898