L(s) = 1 | − 0.326i·3-s + (0.278 − 2.21i)5-s + (−2.25 − 1.38i)7-s + 2.89·9-s + 4.91i·11-s + 4.89·13-s + (−0.724 − 0.0908i)15-s + 6.14·17-s − 4.72·19-s + (−0.452 + 0.735i)21-s + 6.30·23-s + (−4.84 − 1.23i)25-s − 1.92i·27-s − 4.66·29-s + 4.98·31-s + ⋯ |
L(s) = 1 | − 0.188i·3-s + (0.124 − 0.992i)5-s + (−0.852 − 0.523i)7-s + 0.964·9-s + 1.48i·11-s + 1.35·13-s + (−0.187 − 0.0234i)15-s + 1.49·17-s − 1.08·19-s + (−0.0986 + 0.160i)21-s + 1.31·23-s + (−0.969 − 0.247i)25-s − 0.370i·27-s − 0.866·29-s + 0.895·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.988881686\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.988881686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.278 + 2.21i)T \) |
| 7 | \( 1 + (2.25 + 1.38i)T \) |
good | 3 | \( 1 + 0.326iT - 3T^{2} \) |
| 11 | \( 1 - 4.91iT - 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 6.14T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 - 6.30T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 - 4.98T + 31T^{2} \) |
| 37 | \( 1 - 2.88iT - 37T^{2} \) |
| 41 | \( 1 - 7.38iT - 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 47 | \( 1 - 5.09iT - 47T^{2} \) |
| 53 | \( 1 + 1.63iT - 53T^{2} \) |
| 59 | \( 1 + 1.30T + 59T^{2} \) |
| 61 | \( 1 + 8.50iT - 61T^{2} \) |
| 67 | \( 1 + 3.67T + 67T^{2} \) |
| 71 | \( 1 + 1.27iT - 71T^{2} \) |
| 73 | \( 1 - 8.00T + 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 - 0.484iT - 83T^{2} \) |
| 89 | \( 1 + 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−9.108415852704022057440393382328, −8.089132083128357094263611174817, −7.44211872734528485961842020238, −6.62389290954277540986903925578, −5.92428540525432351189723986341, −4.75296228401814183819639517980, −4.22346954883301940947763114959, −3.25799842012287710454063182019, −1.73782876554067853499155825065, −0.956956972417776850705505836504,
1.01015943888241378883689503292, 2.51192160365486445391592726903, 3.48703192738941515511785740770, 3.82564370829889932843163353751, 5.45230818488984889622141340956, 6.02807976796352523308407448617, 6.66476207854274178005621277687, 7.48406020848216537253400658222, 8.471979667881509392637899982090, 9.111001817873345638177488874110