L(s) = 1 | + (1.66 − 2.87i)5-s − 2.71i·7-s + 0.985i·11-s + (4.33 − 2.50i)13-s + (2.51 − 4.35i)17-s + (0.193 − 4.35i)19-s + (−3.68 + 2.12i)23-s + (−3.01 − 5.22i)25-s + (−5.83 + 3.36i)29-s + 2.32·31-s + (−7.81 − 4.51i)35-s − 8.27i·37-s + (9.96 + 5.75i)41-s + (9.48 + 5.47i)43-s + (−6.41 + 3.70i)47-s + ⋯ |
L(s) = 1 | + (0.742 − 1.28i)5-s − 1.02i·7-s + 0.297i·11-s + (1.20 − 0.694i)13-s + (0.609 − 1.05i)17-s + (0.0443 − 0.999i)19-s + (−0.769 + 0.444i)23-s + (−0.602 − 1.04i)25-s + (−1.08 + 0.625i)29-s + 0.416·31-s + (−1.32 − 0.762i)35-s − 1.36i·37-s + (1.55 + 0.898i)41-s + (1.44 + 0.834i)43-s + (−0.935 + 0.540i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.143908199\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.143908199\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.193 + 4.35i)T \) |
good | 5 | \( 1 + (-1.66 + 2.87i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 2.71iT - 7T^{2} \) |
| 11 | \( 1 - 0.985iT - 11T^{2} \) |
| 13 | \( 1 + (-4.33 + 2.50i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.51 + 4.35i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (3.68 - 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.83 - 3.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 + 8.27iT - 37T^{2} \) |
| 41 | \( 1 + (-9.96 - 5.75i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.48 - 5.47i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.41 - 3.70i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.14 - 2.97i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.66 + 2.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.62 - 6.28i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.67 + 11.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.19 - 3.79i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.52 - 4.37i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.48 - 9.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.70iT - 83T^{2} \) |
| 89 | \( 1 + (-6.10 + 3.52i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.12 + 4.11i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.658262926341398638386760552607, −7.78464547981729923823585021630, −7.23731749366624570465538834859, −6.05900445859745293351422239214, −5.51841242559288452138695455753, −4.64555992222260950803125042190, −3.94564731283392937911681530454, −2.79410428810237410622810586999, −1.41605480444089600546250177525, −0.73328609976646192169743460525,
1.64142948456804002362074995819, 2.37724053085598144865866202005, 3.38801947564209862461146589452, 4.11018621244616185360773034204, 5.68745934300661479593937442698, 5.97892576288560148017210586640, 6.49209961274152655839601198182, 7.58980929935333616703743706190, 8.367600803103527153950135485883, 9.049253531531749578710972286313