| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.198 − 0.171i)3-s + (0.841 + 0.540i)4-s + (−0.382 + 2.66i)5-s + (0.141 + 0.220i)6-s + (0.921 − 2.47i)7-s + (−0.654 − 0.755i)8-s + (−0.417 − 2.90i)9-s + (1.11 − 2.44i)10-s + (1.12 + 3.81i)11-s + (−0.0739 − 0.251i)12-s + (4.25 + 1.94i)13-s + (−1.58 + 2.11i)14-s + (0.533 − 0.462i)15-s + (0.415 + 0.909i)16-s + (−5.41 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.114 − 0.0992i)3-s + (0.420 + 0.270i)4-s + (−0.171 + 1.19i)5-s + (0.0579 + 0.0901i)6-s + (0.348 − 0.937i)7-s + (−0.231 − 0.267i)8-s + (−0.139 − 0.967i)9-s + (0.353 − 0.773i)10-s + (0.337 + 1.15i)11-s + (−0.0213 − 0.0727i)12-s + (1.17 + 0.538i)13-s + (−0.423 + 0.566i)14-s + (0.137 − 0.119i)15-s + (0.103 + 0.227i)16-s + (−1.31 + 0.843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.940318 + 0.218026i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.940318 + 0.218026i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (-0.921 + 2.47i)T \) |
| 23 | \( 1 + (-4.50 - 1.63i)T \) |
| good | 3 | \( 1 + (0.198 + 0.171i)T + (0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.382 - 2.66i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-1.12 - 3.81i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-4.25 - 1.94i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.41 - 3.47i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-5.37 - 3.45i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-8.32 + 5.34i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.278 + 0.241i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (5.74 - 0.825i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.832 - 0.119i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-1.45 - 1.26i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.02iT - 47T^{2} \) |
| 53 | \( 1 + (8.17 - 3.73i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.80 - 0.824i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (4.61 + 5.32i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.06 - 3.61i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-1.23 - 0.362i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-7.80 + 12.1i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.00 - 0.460i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (2.11 + 14.7i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (2.64 - 3.05i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 16.6i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.42012342965321915364308404918, −10.78979538436980618826839827694, −9.955133871486136458897964555272, −8.989543211751150441672305804180, −7.79793336657586624652309136730, −6.82580169023398982971211015826, −6.39565817815205808459278814950, −4.26266138047179597537846562741, −3.27339128771701508541105937755, −1.48410969011508267395815110525,
1.04573861393599405586888060635, 2.86558953218712796068199180478, 4.87610470579992803230692669419, 5.45981538055184976842643216927, 6.76201127759052290634544816225, 8.261555197795566187091169435810, 8.641752718226162126168544051433, 9.274779247256048879311783494450, 10.89522270672300288076695088027, 11.26369599676578990184826742252
