| L(s) = 1 | + (−0.938 + 2.84i)3-s + (−4.88 − 1.05i)5-s + 6.81i·7-s + (−7.23 − 5.34i)9-s − 7.52i·11-s − 16.2i·13-s + (7.58 − 12.9i)15-s − 4.11·17-s − 7.86·19-s + (−19.4 − 6.39i)21-s − 19.5·23-s + (22.7 + 10.2i)25-s + (22.0 − 15.6i)27-s − 55.8i·29-s − 43.4·31-s + ⋯ |
| L(s) = 1 | + (−0.312 + 0.949i)3-s + (−0.977 − 0.210i)5-s + 0.973i·7-s + (−0.804 − 0.594i)9-s − 0.684i·11-s − 1.24i·13-s + (0.505 − 0.862i)15-s − 0.242·17-s − 0.413·19-s + (−0.924 − 0.304i)21-s − 0.848·23-s + (0.911 + 0.411i)25-s + (0.815 − 0.578i)27-s − 1.92i·29-s − 1.40·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0829951 - 0.144830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0829951 - 0.144830i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.938 - 2.84i)T \) |
| 5 | \( 1 + (4.88 + 1.05i)T \) |
| good | 7 | \( 1 - 6.81iT - 49T^{2} \) |
| 11 | \( 1 + 7.52iT - 121T^{2} \) |
| 13 | \( 1 + 16.2iT - 169T^{2} \) |
| 17 | \( 1 + 4.11T + 289T^{2} \) |
| 19 | \( 1 + 7.86T + 361T^{2} \) |
| 23 | \( 1 + 19.5T + 529T^{2} \) |
| 29 | \( 1 + 55.8iT - 841T^{2} \) |
| 31 | \( 1 + 43.4T + 961T^{2} \) |
| 37 | \( 1 - 31.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 78.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.72T + 6.88e3T^{2} \) |
| 89 | \( 1 - 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44iT - 9.40e3T^{2} \) |
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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
−11.55568958195088464265906298762, −10.74820121648690163699812083756, −9.672749068484966263193380892212, −8.595229407002906612878900068637, −7.964410474278609635622872181453, −6.17608038248609443742631091324, −5.30576213471870479352553976973, −4.08931521504574211871710363030, −2.95519425305292711617481044833, −0.089767550623255913970475763361,
1.78222536992214321262451007784, 3.68772097564330456928486313712, 4.84241693708613125158216523153, 6.58185488801996224868371203475, 7.16293800436035523365284498607, 7.984378780896775543073784400751, 9.195573657589996650929738624720, 10.71194463462243889743171183654, 11.25561288893850961263658525693, 12.30490416841092409143766239625
