| 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
−12.25676501475320928067217075787, −11.32532907366063731740926390493, −10.74317153062558871829048264394, −9.775761154714727093003522052521, −8.975157679738049347141512914538, −7.65223401105433505267041455522, −6.08887814513098129837853237004, −5.20327295218478310746168997571, −3.40683820826588175345432816414, −2.26576693511251838110254722210,
1.58587727083958058526286202637, 3.36114280654995235000142256048, 5.18819969642669466679820691689, 6.70624061941574819423462279018, 7.25137584709296780813868934743, 8.337597743475535355002663913362, 9.016909214754201420574288977759, 10.78074066656550957094586380997, 11.69394956368943803296575233964, 12.53244786418508093662050649688
