| L(s) = 1 | + (1.31 − 2.69i)3-s + (−1.53 − 4.75i)5-s + 12.2i·7-s + (−5.56 − 7.07i)9-s − 3.02·11-s + 13.7·13-s + (−14.8 − 2.08i)15-s − 25.8·17-s + 14.0i·19-s + (33.0 + 16.0i)21-s − 17.8·23-s + (−20.2 + 14.6i)25-s + (−26.3 + 5.72i)27-s − 27.5·29-s + 33.8·31-s + ⋯ |
| L(s) = 1 | + (0.437 − 0.899i)3-s + (−0.307 − 0.951i)5-s + 1.74i·7-s + (−0.617 − 0.786i)9-s − 0.274·11-s + 1.05·13-s + (−0.990 − 0.139i)15-s − 1.52·17-s + 0.741i·19-s + (1.57 + 0.764i)21-s − 0.775·23-s + (−0.810 + 0.585i)25-s + (−0.977 + 0.211i)27-s − 0.951·29-s + 1.09·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6476069617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6476069617\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.31 + 2.69i)T \) |
| 5 | \( 1 + (1.53 + 4.75i)T \) |
| good | 7 | \( 1 - 12.2iT - 49T^{2} \) |
| 11 | \( 1 + 3.02T + 121T^{2} \) |
| 13 | \( 1 - 13.7T + 169T^{2} \) |
| 17 | \( 1 + 25.8T + 289T^{2} \) |
| 19 | \( 1 - 14.0iT - 361T^{2} \) |
| 23 | \( 1 + 17.8T + 529T^{2} \) |
| 29 | \( 1 + 27.5T + 841T^{2} \) |
| 31 | \( 1 - 33.8T + 961T^{2} \) |
| 37 | \( 1 + 14.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 63.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 28.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 2.88iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 69.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 69.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 113. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 109.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 16.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 66.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 62.9iT - 9.40e3T^{2} \) |
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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
−9.700341316254520507638238087410, −8.901629850281817932644203148691, −8.432181655587327916247402421909, −7.902473691469570681132975600262, −6.44958790855782927160266661053, −5.93289623208926392468345049856, −4.91610555529629125836547349285, −3.62451705911049581860149065628, −2.39340267417290793909865782359, −1.51867980639297151864815479731,
0.18441501476019848557508224446, 2.20053744547963024794203828352, 3.56210874675942387980959407703, 3.93806988507762342313944153012, 4.93010770071161369329396645817, 6.39510748635053172629646551983, 7.06398807037817989407117092045, 7.978627457115212212545675846984, 8.734086334226980788798325167338, 9.885998133123492321359739068660
