L(s) = 1 | + (1.70 + 0.292i)3-s + (−3 + 3i)7-s + (2.82 + i)9-s + 1.41i·11-s + (4.24 + 4.24i)17-s + 4i·19-s + (−5.99 + 4.24i)21-s + (2.82 − 2.82i)23-s + (4.53 + 2.53i)27-s − 1.41·29-s − 2·31-s + (−0.414 + 2.41i)33-s + (2 − 2i)37-s − 5.65i·41-s + (2 + 2i)43-s + ⋯ |
L(s) = 1 | + (0.985 + 0.169i)3-s + (−1.13 + 1.13i)7-s + (0.942 + 0.333i)9-s + 0.426i·11-s + (1.02 + 1.02i)17-s + 0.917i·19-s + (−1.30 + 0.925i)21-s + (0.589 − 0.589i)23-s + (0.872 + 0.487i)27-s − 0.262·29-s − 0.359·31-s + (−0.0721 + 0.420i)33-s + (0.328 − 0.328i)37-s − 0.883i·41-s + (0.304 + 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52005 + 1.00575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52005 + 1.00575i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 - 0.292i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 2i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2 - 2i)T + 43iT^{2} \) |
| 47 | \( 1 + (5.65 + 5.65i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (4 - 4i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 + (2.82 - 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-13 + 13i)T - 97iT^{2} \) |
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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
−10.44374373943198882653260644362, −9.877114515604643061740240731342, −9.061098526170199571087123672546, −8.379170520530547041091354849744, −7.39594522252576426712038542949, −6.31798232045585439324785017930, −5.37721151937370020053293826708, −3.91712875277850756595439968011, −3.07992668812418936133010503143, −1.94767599230010331403798451430,
0.964903326611418223531473243068, 2.88628410245204732300949769799, 3.50786974838379001124111800502, 4.69358348952496015804781533569, 6.18200349647624083788691126250, 7.21819065090895601978676571231, 7.60689976931494954792452632750, 8.928954775084505477737826154369, 9.566793945218406773050872875075, 10.25704470678019002924901755503