L(s) = 1 | + (1.93 + 1.11i)2-s + (1.5 + 2.59i)4-s + (1.93 − 1.11i)5-s + 2.23i·8-s + 5.00·10-s + (0.499 − 0.866i)16-s + 4.47i·17-s − 4·19-s + (5.80 + 3.35i)20-s + (−7.74 + 4.47i)23-s + (2.5 − 4.33i)25-s + (−4 − 6.92i)31-s + (5.80 − 3.35i)32-s + (−5.00 + 8.66i)34-s + (−7.74 − 4.47i)38-s + ⋯ |
L(s) = 1 | + (1.36 + 0.790i)2-s + (0.750 + 1.29i)4-s + (0.866 − 0.499i)5-s + 0.790i·8-s + 1.58·10-s + (0.124 − 0.216i)16-s + 1.08i·17-s − 0.917·19-s + (1.29 + 0.750i)20-s + (−1.61 + 0.932i)23-s + (0.5 − 0.866i)25-s + (−0.718 − 1.24i)31-s + (1.02 − 0.592i)32-s + (−0.857 + 1.48i)34-s + (−1.25 − 0.725i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.75530 + 1.28481i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.75530 + 1.28481i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
good | 2 | \( 1 + (-1.93 - 1.11i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (7.74 - 4.47i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.74 - 4.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 4.47iT - 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (15.4 + 8.94i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (48.5 + 84.0i)T^{2} \) |
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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
−11.78159301932611246525955974227, −10.48315337185775392594177960285, −9.557782904022182929610821154323, −8.398389830964368102909002806880, −7.41347174060500359672031205677, −6.06176014302803497260194533397, −5.88884405147966952323756211855, −4.61365207986937693365452577404, −3.74433880585765858894579218498, −2.06485102162863990321339208821,
1.95545951701561982223894783375, 2.87834608732151147481978912402, 4.10498306318368326979368505879, 5.18010291709208538409372949525, 6.05964597936380471116148728792, 6.95436069193290696773683620165, 8.476138998771437816923039085107, 9.716881328814181500310617100251, 10.52756710895969887080064607474, 11.19113092757115705953356890090