L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (3.15 − 1.82i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.44 − 0.834i)11-s + 0.999·12-s + (2.24 − 2.82i)13-s − 3.64·14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + 0.999i·18-s + (5.46 − 3.15i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (1.19 − 0.688i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.435 − 0.251i)11-s + 0.288·12-s + (0.622 − 0.782i)13-s − 0.974·14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + 0.235i·18-s + (1.25 − 0.724i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.502367512\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.502367512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.24 + 2.82i)T \) |
good | 7 | \( 1 + (-3.15 + 1.82i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.44 + 0.834i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.46 + 3.15i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.622 - 1.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.02 - 8.69i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.21iT - 31T^{2} \) |
| 37 | \( 1 + (8.54 + 4.93i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.04 - 4.64i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.78 - 6.55i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6.82iT - 47T^{2} \) |
| 53 | \( 1 - 0.848T + 53T^{2} \) |
| 59 | \( 1 + (-5.29 + 3.05i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.73 + 6.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.7 + 7.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.04 - 1.75i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 12.2iT - 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 - 7.95iT - 83T^{2} \) |
| 89 | \( 1 + (-5.15 - 2.97i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.38 + 1.37i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.895794365883946125778836138435, −8.041890368374721685476137935120, −7.53029462220778249475719741090, −6.97112550175212764982394284552, −5.65184048759144130378573884155, −4.87530748452526900328553960607, −3.66307500441253365764896145994, −2.78270509646176347915409655739, −1.61795694135270102328332582110, −0.67087426313688332509612263100,
1.53304504549704020703507797225, 2.33892474589656356350513027110, 3.75379232777791762799077584420, 4.67332241429922572882266911231, 5.53859122970655456519663717931, 6.19726877598266817079751797589, 7.48762255885885951832182350205, 7.916642670849023062089905833517, 8.896905933568666827312437277268, 9.060291965314153520074690543989