L(s) = 1 | − 3.11·3-s + (−0.660 − 2.13i)5-s + i·7-s + 6.70·9-s + 0.780i·11-s − 3.09·13-s + (2.05 + 6.65i)15-s − 7.31i·17-s + 0.207i·19-s − 3.11i·21-s + 1.52i·23-s + (−4.12 + 2.82i)25-s − 11.5·27-s − 3.60i·29-s + 7.94·31-s + ⋯ |
L(s) = 1 | − 1.79·3-s + (−0.295 − 0.955i)5-s + 0.377i·7-s + 2.23·9-s + 0.235i·11-s − 0.857·13-s + (0.531 + 1.71i)15-s − 1.77i·17-s + 0.0476i·19-s − 0.679i·21-s + 0.318i·23-s + (−0.825 + 0.564i)25-s − 2.22·27-s − 0.669i·29-s + 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2850956857\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2850956857\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.660 + 2.13i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 11 | \( 1 - 0.780iT - 11T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 17 | \( 1 + 7.31iT - 17T^{2} \) |
| 19 | \( 1 - 0.207iT - 19T^{2} \) |
| 23 | \( 1 - 1.52iT - 23T^{2} \) |
| 29 | \( 1 + 3.60iT - 29T^{2} \) |
| 31 | \( 1 - 7.94T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 - 6.61T + 43T^{2} \) |
| 47 | \( 1 + 2.61iT - 47T^{2} \) |
| 53 | \( 1 - 8.69T + 53T^{2} \) |
| 59 | \( 1 + 14.6iT - 59T^{2} \) |
| 61 | \( 1 - 6.37iT - 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 0.502iT - 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 10.2iT - 97T^{2} \) |
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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
−8.754048763512886463997532034177, −7.57468154521400302213397660665, −7.13801215415009037854904834606, −6.04707467835126636038522122190, −5.46858310894298508727011349733, −4.72119249714995998903928571948, −4.30932632734395662389177964087, −2.57807312660489786887988732442, −1.09842752836865938764660086924, −0.16296948361691683230271853788,
1.22478932008682062769819727777, 2.70446576603150393460662775488, 4.08137892011794243190238629981, 4.53762163455599352625674388362, 5.85197465693249517390688065189, 6.07114218132720560653194967761, 7.03663525852036126074651933238, 7.48042011057528116887475037219, 8.542595505338610799460758494222, 9.912892831552682802560226449597