| L(s) = 1 | − 1.73i·3-s − 0.816·5-s + 6.23·7-s − 2.99·9-s − 7.41·11-s − 0.641i·13-s + 1.41i·15-s − 26.4·17-s + (−9.92 + 16.2i)19-s − 10.7i·21-s + 14.1·23-s − 24.3·25-s + 5.19i·27-s − 45.5i·29-s + 24.8i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.163·5-s + 0.890·7-s − 0.333·9-s − 0.674·11-s − 0.0493i·13-s + 0.0942i·15-s − 1.55·17-s + (−0.522 + 0.852i)19-s − 0.514i·21-s + 0.615·23-s − 0.973·25-s + 0.192i·27-s − 1.57i·29-s + 0.801i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03753806816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03753806816\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 19 | \( 1 + (9.92 - 16.2i)T \) |
| good | 5 | \( 1 + 0.816T + 25T^{2} \) |
| 7 | \( 1 - 6.23T + 49T^{2} \) |
| 11 | \( 1 + 7.41T + 121T^{2} \) |
| 13 | \( 1 + 0.641iT - 169T^{2} \) |
| 17 | \( 1 + 26.4T + 289T^{2} \) |
| 23 | \( 1 - 14.1T + 529T^{2} \) |
| 29 | \( 1 + 45.5iT - 841T^{2} \) |
| 31 | \( 1 - 24.8iT - 961T^{2} \) |
| 37 | \( 1 - 15.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 23.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 47.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 89.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 16.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 4.08iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 68.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 34.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 11.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 150. iT - 9.40e3T^{2} \) |
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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
−10.38996614843987579642271814463, −9.300412568438294266485465711994, −8.268317964210788149678888775077, −7.917431384699192422506042348452, −6.87151911170859319668159279483, −5.99342095125171580493423042151, −4.96124516759358562982029195472, −4.05977096808597976237410711930, −2.58447454848957598055373640433, −1.63639807801175637233273439959,
0.01128806493934972692463945482, 1.89270815014483314273355971869, 3.04768829464342308660030366560, 4.42944619138680775190167110647, 4.84427586608271011381951659532, 5.98507020547513087952698939756, 7.06556960386238116563565858907, 7.970176824173268766162810224445, 8.785527155010893862555064225030, 9.430725621229504315134781169268
