| L(s) = 1 | + (−1.24 + 1.78i)2-s + (−1.73 + 0.0839i)3-s + (−0.936 − 2.57i)4-s + (−0.952 + 0.0833i)5-s + (2.01 − 3.18i)6-s + (−1.93 − 1.62i)7-s + (1.55 + 0.415i)8-s + (2.98 − 0.290i)9-s + (1.04 − 1.80i)10-s + (−1.42 − 2.47i)11-s + (1.83 + 4.37i)12-s + (0.354 + 0.165i)13-s + (5.32 − 1.42i)14-s + (1.64 − 0.224i)15-s + (1.51 − 1.27i)16-s + (−5.70 + 2.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.882 + 1.26i)2-s + (−0.998 + 0.0484i)3-s + (−0.468 − 1.28i)4-s + (−0.426 + 0.0372i)5-s + (0.820 − 1.30i)6-s + (−0.733 − 0.615i)7-s + (0.548 + 0.147i)8-s + (0.995 − 0.0967i)9-s + (0.329 − 0.570i)10-s + (−0.430 − 0.746i)11-s + (0.530 + 1.26i)12-s + (0.0982 + 0.0458i)13-s + (1.42 − 0.381i)14-s + (0.423 − 0.0578i)15-s + (0.378 − 0.317i)16-s + (−1.38 + 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0544 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0544 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0313722 - 0.0331285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0313722 - 0.0331285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 - 0.0839i)T \) |
| 37 | \( 1 + (-2.35 - 5.60i)T \) |
| good | 2 | \( 1 + (1.24 - 1.78i)T + (-0.684 - 1.87i)T^{2} \) |
| 5 | \( 1 + (0.952 - 0.0833i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (1.93 + 1.62i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (1.42 + 2.47i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.354 - 0.165i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (5.70 - 2.66i)T + (10.9 - 13.0i)T^{2} \) |
| 19 | \( 1 + (4.77 - 3.34i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (1.49 + 5.57i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.367 - 1.37i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (0.244 - 0.244i)T - 31iT^{2} \) |
| 41 | \( 1 + (-9.64 + 3.51i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.04 + 2.04i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.12 - 4.68i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.85 + 4.59i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.227 + 2.59i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (5.29 - 11.3i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (-1.63 + 1.95i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (13.4 + 2.36i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 11.5iT - 73T^{2} \) |
| 79 | \( 1 + (0.610 + 6.97i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (5.43 - 14.9i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.22 - 0.282i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (-5.15 + 1.38i)T + (84.0 - 48.5i)T^{2} \) |
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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
−13.40974745467534135590112588232, −12.36986957076774022080656213645, −10.90103406515492681010209634273, −10.20539377074064689232195672771, −8.844043829721323093041525373843, −7.74485737543301896853233942379, −6.57719466501392968517524356432, −5.98152292376386846251003569735, −4.18890302872479460013780522852, −0.07170931306931781221177903935,
2.34091213805623036812095359820, 4.28572145972071037939377862392, 6.03572725984781204461216055080, 7.47301725667388286782723432261, 9.036002937574367008617745085585, 9.822845894677571217349012540358, 10.94432951756626828596489272638, 11.57716946084135532502933912105, 12.54377706214605961699505199699, 13.16834020501380992357451084222
