| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.326 + 1.85i)3-s + (0.766 + 0.642i)4-s + (1.53 − 1.28i)5-s + (−0.326 + 1.85i)6-s + (−2.53 + 4.38i)7-s + (0.500 + 0.866i)8-s + (−0.5 + 0.181i)9-s + (1.87 − 0.684i)10-s + (0.705 + 1.22i)11-s + (−0.939 + 1.62i)12-s + (0.226 − 1.28i)13-s + (−3.87 + 3.25i)14-s + (2.87 + 2.41i)15-s + (0.173 + 0.984i)16-s + (−2.24 − 0.817i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.188 + 1.06i)3-s + (0.383 + 0.321i)4-s + (0.685 − 0.574i)5-s + (−0.133 + 0.755i)6-s + (−0.957 + 1.65i)7-s + (0.176 + 0.306i)8-s + (−0.166 + 0.0606i)9-s + (0.594 − 0.216i)10-s + (0.212 + 0.368i)11-s + (−0.271 + 0.469i)12-s + (0.0628 − 0.356i)13-s + (−1.03 + 0.869i)14-s + (0.743 + 0.623i)15-s + (0.0434 + 0.246i)16-s + (−0.544 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33392 + 2.01921i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33392 + 2.01921i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-0.326 - 1.85i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.53 + 1.28i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (2.53 - 4.38i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.705 - 1.22i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.226 + 1.28i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.24 + 0.817i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.34 + 1.96i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.94 + 2.89i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.184 + 0.320i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + (-0.266 - 1.50i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.581 - 0.487i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.59 + 3.49i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.28 + 1.07i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.673 - 0.245i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.47 - 6.27i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.31 - 0.480i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.87 + 4.09i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.791 + 4.49i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.389 - 2.20i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.99 + 3.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.84 + 10.4i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.43 + 0.524i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.42084326296223072342546494370, −9.731581442619230919791491989079, −9.051713216003896039531363320874, −8.452916981497610949745697588954, −6.83969470139087646809697550511, −5.96365151949175398527590255708, −5.26380958699408767678311106320, −4.39991105498692192660518615128, −3.22154758886822810932585796141, −2.23300164400365829430632797086,
1.04098219479655844025625102162, 2.32288552666733535725052425305, 3.46565050364324814753891095123, 4.41718586429100400361218531273, 5.99665246967844206311350422180, 6.78589344704111339615090315686, 6.99747814447100580821537135514, 8.173290216116285527471070377334, 9.547414132846342820168882243058, 10.35988573621690330452739280527
