L(s) = 1 | + (0.756 + 0.756i)3-s + (−8.22 + 8.22i)5-s + 2.67i·7-s − 25.8i·9-s + (−45.2 + 45.2i)11-s + (−35.3 − 35.3i)13-s − 12.4·15-s − 72.4·17-s + (19.4 + 19.4i)19-s + (−2.02 + 2.02i)21-s + 139. i·23-s − 10.3i·25-s + (39.9 − 39.9i)27-s + (−66.0 − 66.0i)29-s − 188.·31-s + ⋯ |
L(s) = 1 | + (0.145 + 0.145i)3-s + (−0.735 + 0.735i)5-s + 0.144i·7-s − 0.957i·9-s + (−1.23 + 1.23i)11-s + (−0.755 − 0.755i)13-s − 0.214·15-s − 1.03·17-s + (0.234 + 0.234i)19-s + (−0.0210 + 0.0210i)21-s + 1.26i·23-s − 0.0826i·25-s + (0.285 − 0.285i)27-s + (−0.422 − 0.422i)29-s − 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0487301 + 0.423754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0487301 + 0.423754i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.756 - 0.756i)T + 27iT^{2} \) |
| 5 | \( 1 + (8.22 - 8.22i)T - 125iT^{2} \) |
| 7 | \( 1 - 2.67iT - 343T^{2} \) |
| 11 | \( 1 + (45.2 - 45.2i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (35.3 + 35.3i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 + 72.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-19.4 - 19.4i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 139. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (66.0 + 66.0i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-84.0 + 84.0i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 104. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (31.4 - 31.4i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (149. - 149. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-284. + 284. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-228. - 228. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (-139. - 139. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 - 453. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 259. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (563. + 563. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 866. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 936.T + 9.12e5T^{2} \) |
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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
−13.17362659885731773536062748209, −12.32809510812336779233916695471, −11.28754788965017974045406740374, −10.22541608651181725071865569902, −9.280641541110581441191334748928, −7.72534737564448214716709650415, −7.10486034231696241376949611967, −5.44705066114471587000393800643, −3.94281970347378143155949022699, −2.58312204320429630935646757056,
0.20163518818049323043374773007, 2.47670928151610084877061776029, 4.32891957454080549906193635195, 5.37906182537608873431382203400, 7.13079273697180309411543776088, 8.170947221108921189667381993949, 8.908607693648324955627992723705, 10.53739808471319104377496886269, 11.29372681905868675571576648183, 12.52557396861788923491531625709