L(s) = 1 | + (1.22 − 2.12i)5-s + (0.5 + 2.59i)7-s + (−1.22 + 0.707i)11-s + 5.19i·13-s + (2.44 + 4.24i)17-s + (−1.5 − 0.866i)19-s + (4.89 + 2.82i)23-s + (−0.499 − 0.866i)25-s − 2.82i·29-s + (−1.5 + 0.866i)31-s + (6.12 + 2.12i)35-s + (0.5 − 0.866i)37-s + 7.34·41-s + 43-s + (6.12 − 10.6i)47-s + ⋯ |
L(s) = 1 | + (0.547 − 0.948i)5-s + (0.188 + 0.981i)7-s + (−0.369 + 0.213i)11-s + 1.44i·13-s + (0.594 + 1.02i)17-s + (−0.344 − 0.198i)19-s + (1.02 + 0.589i)23-s + (−0.0999 − 0.173i)25-s − 0.525i·29-s + (−0.269 + 0.155i)31-s + (1.03 + 0.358i)35-s + (0.0821 − 0.142i)37-s + 1.14·41-s + 0.152·43-s + (0.893 − 1.54i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.726653302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.726653302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.34T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-6.12 + 10.6i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 - 1.41i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.44 + 4.24i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + (2.44 - 4.24i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−9.819877352769705791986382789942, −9.099596471745884191839187452120, −8.682292580276857373677941985082, −7.65389901883267895571344510547, −6.53264035431712112126521028713, −5.62625932059164759360601788020, −4.98075336179054579389566988502, −3.93858783417101216152425231948, −2.40036120951079668965161460458, −1.46807278329741051670278782812,
0.861898717497894710039953335092, 2.63135504273872976414255921917, 3.31402426335908474762141077242, 4.66543206211386908994888247897, 5.61836615368041125296047072665, 6.53210884034060517273424955987, 7.42643064060186659934929097205, 7.948596937917136055014822951312, 9.199759601339054656940399545594, 10.07146231123935133526900293759