| L(s) = 1 | + (−0.234 + 0.645i)2-s + (0.440 − 1.67i)3-s + (1.17 + 0.982i)4-s + (0.231 − 1.31i)5-s + (0.977 + 0.677i)6-s + (1.38 − 2.25i)7-s + (−2.09 + 1.21i)8-s + (−2.61 − 1.47i)9-s + (0.793 + 0.458i)10-s + (0.216 − 0.0381i)11-s + (2.16 − 1.52i)12-s + (0.673 + 1.85i)13-s + (1.13 + 1.42i)14-s + (−2.10 − 0.967i)15-s + (0.242 + 1.37i)16-s + (−0.469 + 0.813i)17-s + ⋯ |
| L(s) = 1 | + (−0.166 + 0.456i)2-s + (0.254 − 0.967i)3-s + (0.585 + 0.491i)4-s + (0.103 − 0.588i)5-s + (0.398 + 0.276i)6-s + (0.521 − 0.852i)7-s + (−0.741 + 0.428i)8-s + (−0.870 − 0.492i)9-s + (0.251 + 0.144i)10-s + (0.0652 − 0.0115i)11-s + (0.624 − 0.441i)12-s + (0.186 + 0.513i)13-s + (0.302 + 0.379i)14-s + (−0.542 − 0.249i)15-s + (0.0605 + 0.343i)16-s + (−0.113 + 0.197i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31818 - 0.260479i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31818 - 0.260479i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.440 + 1.67i)T \) |
| 7 | \( 1 + (-1.38 + 2.25i)T \) |
| good | 2 | \( 1 + (0.234 - 0.645i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.231 + 1.31i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.216 + 0.0381i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.673 - 1.85i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.469 - 0.813i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.04 + 1.75i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.17 + 1.40i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (1.33 - 3.66i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (5.95 - 7.09i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.56 - 2.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (10.8 - 3.95i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.261 + 1.48i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.18 + 4.35i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 3.77iT - 53T^{2} \) |
| 59 | \( 1 + (1.72 - 9.76i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (6.04 + 7.20i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (6.36 - 2.31i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-11.8 - 6.85i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.68 + 4.43i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.91 + 2.88i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-16.1 - 5.88i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (3.46 + 5.99i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.42 - 0.780i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.53244786418508093662050649688, −11.69394956368943803296575233964, −10.78074066656550957094586380997, −9.016909214754201420574288977759, −8.337597743475535355002663913362, −7.25137584709296780813868934743, −6.70624061941574819423462279018, −5.18819969642669466679820691689, −3.36114280654995235000142256048, −1.58587727083958058526286202637,
2.26576693511251838110254722210, 3.40683820826588175345432816414, 5.20327295218478310746168997571, 6.08887814513098129837853237004, 7.65223401105433505267041455522, 8.975157679738049347141512914538, 9.775761154714727093003522052521, 10.74317153062558871829048264394, 11.32532907366063731740926390493, 12.25676501475320928067217075787
