| L(s) = 1 | − 2.04·2-s + (−1.22 − 1.22i)3-s + 2.18·4-s + i·5-s + (2.50 + 2.50i)6-s − i·7-s − 0.377·8-s + (−0.00153 + 2.99i)9-s − 2.04i·10-s + (−0.566 + 3.26i)11-s + (−2.67 − 2.67i)12-s − 2.37i·13-s + 2.04i·14-s + (1.22 − 1.22i)15-s − 3.59·16-s − 4.73·17-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + (−0.706 − 0.707i)3-s + 1.09·4-s + 0.447i·5-s + (1.02 + 1.02i)6-s − 0.377i·7-s − 0.133·8-s + (−0.000510 + 0.999i)9-s − 0.646i·10-s + (−0.170 + 0.985i)11-s + (−0.772 − 0.772i)12-s − 0.658i·13-s + 0.546i·14-s + (0.316 − 0.316i)15-s − 0.899·16-s − 1.14·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2013983335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2013983335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.566 - 3.26i)T \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 13 | \( 1 + 2.37iT - 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 + 0.636iT - 19T^{2} \) |
| 23 | \( 1 - 4.83iT - 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 5.92T + 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 6.88iT - 53T^{2} \) |
| 59 | \( 1 - 1.96iT - 59T^{2} \) |
| 61 | \( 1 - 3.95iT - 61T^{2} \) |
| 67 | \( 1 - 0.542T + 67T^{2} \) |
| 71 | \( 1 + 6.29iT - 71T^{2} \) |
| 73 | \( 1 + 6.85iT - 73T^{2} \) |
| 79 | \( 1 - 6.24iT - 79T^{2} \) |
| 83 | \( 1 - 2.16T + 83T^{2} \) |
| 89 | \( 1 + 12.4iT - 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.574923340679114165511838039977, −8.533185664998552668488033280711, −7.78993841188850916110255033687, −7.07683583685084511996076009831, −6.64404394096538869689423386590, −5.38236922017189537485854462024, −4.32965987665435901968957230832, −2.55280054260797696721480318830, −1.55826319450756030732763168265, −0.18721020363918426467746170492,
1.09536805798863153412907371845, 2.65016164956831996106467446052, 4.22187371078525955399153209729, 4.94852639678138931706263460931, 6.21834642152674765989247486895, 6.70658633604668686265744328022, 8.143977608348661035167359699611, 8.591537603020639742381415114308, 9.370596438983455446515594260834, 9.893133550589369270297470840846
