L(s) = 1 | + (−2.19 − 0.713i)2-s + (0.279 − 0.384i)3-s + (2.69 + 1.95i)4-s + (−0.887 + 0.644i)6-s − 3.03i·7-s + (−1.80 − 2.48i)8-s + (0.857 + 2.63i)9-s + (0.618 − 1.90i)11-s + (1.50 − 0.489i)12-s + (1.35 − 0.441i)13-s + (−2.16 + 6.66i)14-s + (0.136 + 0.420i)16-s + (1.09 + 1.50i)17-s − 6.40i·18-s + (0.730 − 0.530i)19-s + ⋯ |
L(s) = 1 | + (−1.55 − 0.504i)2-s + (0.161 − 0.221i)3-s + (1.34 + 0.979i)4-s + (−0.362 + 0.263i)6-s − 1.14i·7-s + (−0.639 − 0.880i)8-s + (0.285 + 0.879i)9-s + (0.186 − 0.573i)11-s + (0.434 − 0.141i)12-s + (0.376 − 0.122i)13-s + (−0.578 + 1.78i)14-s + (0.0341 + 0.105i)16-s + (0.265 + 0.365i)17-s − 1.51i·18-s + (0.167 − 0.121i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.498462 - 0.530808i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.498462 - 0.530808i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (2.19 + 0.713i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.279 + 0.384i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 3.03iT - 7T^{2} \) |
| 11 | \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.35 + 0.441i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.09 - 1.50i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.730 + 0.530i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.16 + 1.02i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-3.20 - 2.32i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.21 + 3.78i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-3.63 + 1.18i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.566 + 1.74i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 3.59iT - 43T^{2} \) |
| 47 | \( 1 + (-2.82 + 3.88i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.58 - 7.68i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.28 + 10.1i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.41 + 13.5i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (6.28 + 8.64i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (10.0 + 7.32i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.254 - 0.0827i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (6.93 + 5.03i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.41 - 10.2i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.47 + 4.53i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.85 + 8.05i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.44778933867064517712890808993, −9.627267827910143583193491366762, −8.576301569749579386381638026186, −7.921171830358949618865553937789, −7.31758513905977664050823554160, −6.23760440619764517327845836591, −4.63110650210130474595841963237, −3.31134679401585691267017034318, −1.94481090798091498943819559218, −0.76196441026879066274146465946,
1.28086116453640222572151095141, 2.74302123884510714470608240525, 4.32307877921599315094205450459, 5.83450515057929777088169308573, 6.53874327141397440514800571128, 7.50175274696229543066377139335, 8.466409795117033610680970730525, 9.034195491683389646723950065986, 9.767090155235815445988175398528, 10.30170931232570715200186543909