| L(s) = 1 | + (−2.24 − 0.626i)2-s + (−1.63 + 0.569i)3-s + (2.95 + 1.78i)4-s + (−0.812 + 0.0315i)5-s + (4.03 − 0.255i)6-s + (1.20 + 0.188i)7-s + (−2.32 − 2.46i)8-s + (2.35 − 1.86i)9-s + (1.84 + 0.437i)10-s + (−0.262 − 1.91i)11-s + (−5.84 − 1.23i)12-s + (−1.56 + 2.28i)13-s + (−2.59 − 1.18i)14-s + (1.31 − 0.514i)15-s + (0.466 + 0.885i)16-s + (−3.40 + 1.71i)17-s + ⋯ |
| L(s) = 1 | + (−1.59 − 0.442i)2-s + (−0.944 + 0.328i)3-s + (1.47 + 0.891i)4-s + (−0.363 + 0.0141i)5-s + (1.64 − 0.104i)6-s + (0.456 + 0.0714i)7-s + (−0.821 − 0.870i)8-s + (0.784 − 0.620i)9-s + (0.584 + 0.138i)10-s + (−0.0790 − 0.578i)11-s + (−1.68 − 0.356i)12-s + (−0.435 + 0.634i)13-s + (−0.694 − 0.315i)14-s + (0.338 − 0.132i)15-s + (0.116 + 0.221i)16-s + (−0.826 + 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0129418 + 0.0562259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0129418 + 0.0562259i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.63 - 0.569i)T \) |
| good | 2 | \( 1 + (2.24 + 0.626i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (0.812 - 0.0315i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-1.20 - 0.188i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.262 + 1.91i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (1.56 - 2.28i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (3.40 - 1.71i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (0.0225 + 0.388i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (1.72 - 4.46i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (5.20 + 3.72i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (3.58 - 4.11i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (5.86 + 7.87i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (0.617 - 2.39i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-0.752 - 0.683i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-1.84 - 2.11i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.44 + 8.21i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (0.578 - 0.236i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (-10.0 + 6.04i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (7.98 - 5.70i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-2.59 - 8.68i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (13.5 - 3.20i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (5.21 - 5.11i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (0.514 + 1.99i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (1.84 - 6.16i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (-5.74 - 0.222i)T + (96.7 + 7.51i)T^{2} \) |
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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
−11.74763450611340660329295383841, −11.43478053998375655993058571203, −10.63164978327255907530036115978, −9.678974977424885883485597322157, −8.868382448734379008714154926849, −7.76577544512156289997128968041, −6.84215135929274878347086949825, −5.44693140612956689608090900052, −3.92057128676201735724616797420, −1.82219242949097515913632380926,
0.084169613517225832505128920361, 1.88084239537315563411029374340, 4.56825927094708552702252152488, 5.90396367067859492401292104312, 7.09718527932638553110003738724, 7.61863738806051329886227754531, 8.654867635622152986861773495712, 9.847154240608536578185120846705, 10.59082615720682764693405024214, 11.38979235138088725020361503543
