L(s) = 1 | + (−1.28 − 0.581i)2-s + (1.32 + 1.50i)4-s + (1.41 − 1.41i)7-s + (−0.832 − 2.70i)8-s + 5.29i·11-s + (−3.74 + 3.74i)13-s + (−2.64 + 1.00i)14-s + (−0.5 + 3.96i)16-s − 5.29·19-s + (3.07 − 6.82i)22-s + (−2.82 − 2.82i)23-s + (7 − 2.64i)26-s + (3.99 + 0.250i)28-s + 8i·29-s − 5.29i·31-s + (2.95 − 4.82i)32-s + ⋯ |
L(s) = 1 | + (−0.911 − 0.411i)2-s + (0.661 + 0.750i)4-s + (0.534 − 0.534i)7-s + (−0.294 − 0.955i)8-s + 1.59i·11-s + (−1.03 + 1.03i)13-s + (−0.707 + 0.267i)14-s + (−0.125 + 0.992i)16-s − 1.21·19-s + (0.656 − 1.45i)22-s + (−0.589 − 0.589i)23-s + (1.37 − 0.518i)26-s + (0.754 + 0.0473i)28-s + 1.48i·29-s − 0.950i·31-s + (0.522 − 0.852i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.373107 + 0.442234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.373107 + 0.442234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 + 0.581i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.41 + 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (3.74 - 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 + 5.29iT - 31T^{2} \) |
| 37 | \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + (5.65 + 5.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (7.48 - 7.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + (8.48 - 8.48i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (7.48 - 7.48i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + (-7.48 - 7.48i)T + 97iT^{2} \) |
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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
−10.25510854720646619380139271301, −9.597973014670193609062498557319, −8.803910265184394220238123267977, −7.78822953426084222994796935605, −7.17173325129665491947416638824, −6.45064180737459582132835225424, −4.70627360562551740393710797177, −4.13175350020204815888190646779, −2.46065904888254643850904940548, −1.65384064700263832498012301901,
0.35278120197509409020160721168, 2.02508318469260493208753279996, 3.14484111123190746287260815506, 4.83788522174759211703610056637, 5.77294195348336329864117064760, 6.35675695212523559581516706332, 7.70725006796530063883779506905, 8.155550248307040950301628466229, 8.884871924798721059535596076432, 9.818783840921486666652106556332