| L(s) = 1 | + (0.813 + 1.96i)2-s + (1.03 + 0.690i)3-s + (−1.78 + 1.78i)4-s + (−1.32 + 1.98i)5-s + (−0.515 + 2.58i)6-s + (0.0574 + 0.0860i)7-s + (−1.01 − 0.421i)8-s + (−0.557 − 1.34i)9-s + (−4.98 − 0.990i)10-s + (1.10 − 5.53i)11-s + (−3.06 + 0.610i)12-s + (2.79 + 2.28i)13-s + (−0.122 + 0.182i)14-s + (−2.74 + 1.13i)15-s + 2.69i·16-s + (−3.65 − 1.90i)17-s + ⋯ |
| L(s) = 1 | + (0.575 + 1.38i)2-s + (0.596 + 0.398i)3-s + (−0.890 + 0.890i)4-s + (−0.593 + 0.888i)5-s + (−0.210 + 1.05i)6-s + (0.0217 + 0.0325i)7-s + (−0.360 − 0.149i)8-s + (−0.185 − 0.448i)9-s + (−1.57 − 0.313i)10-s + (0.331 − 1.66i)11-s + (−0.885 + 0.176i)12-s + (0.774 + 0.632i)13-s + (−0.0326 + 0.0488i)14-s + (−0.707 + 0.293i)15-s + 0.673i·16-s + (−0.887 − 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.618324 + 1.64596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.618324 + 1.64596i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-2.79 - 2.28i)T \) |
| 17 | \( 1 + (3.65 + 1.90i)T \) |
| good | 2 | \( 1 + (-0.813 - 1.96i)T + (-1.41 + 1.41i)T^{2} \) |
| 3 | \( 1 + (-1.03 - 0.690i)T + (1.14 + 2.77i)T^{2} \) |
| 5 | \( 1 + (1.32 - 1.98i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.0574 - 0.0860i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.10 + 5.53i)T + (-10.1 - 4.20i)T^{2} \) |
| 19 | \( 1 + (0.203 + 0.491i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.10 - 7.63i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.945 - 4.75i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (2.08 + 10.4i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (1.04 + 5.23i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (1.06 - 0.711i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-2.69 + 1.11i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + 2.00T + 47T^{2} \) |
| 53 | \( 1 + (1.75 - 4.24i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (4.07 + 1.68i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.30 + 6.54i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-7.19 - 7.19i)T + 67iT^{2} \) |
| 71 | \( 1 + (9.54 - 1.89i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (0.830 + 0.555i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (5.81 + 8.69i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (1.04 - 0.431i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 - 2.70iT - 89T^{2} \) |
| 97 | \( 1 + (9.59 + 6.40i)T + (37.1 + 89.6i)T^{2} \) |
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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
−13.26369651574781623203439772658, −11.39741415310862328421668090294, −11.07822236055912599366810000672, −9.194975601143009539659481666658, −8.592984173458935710726365233161, −7.42935488434655155006794311984, −6.55875137554973880210322708586, −5.64820806239442382642107480509, −4.01439722880773052489660889120, −3.27833994950095715811920285623,
1.49315399904088437024189843612, 2.77466622123298188892915934337, 4.24920904057837356161252700963, 4.91461492452326202572403988030, 6.93434013642057727458971402847, 8.196721754906874885397760338203, 8.976608003875496759587795009373, 10.29875502556030638619440816782, 11.04352729044911109124622768975, 12.27323351721593200645651315799
