| L(s) = 1 | + (1.72 + 4.69i)5-s + 13.6i·7-s − 1.59i·11-s − 17.2i·13-s − 21.0·17-s − 11.6·19-s + 24.4·23-s + (−19.0 + 16.1i)25-s + 40.5i·29-s + 38.6·31-s + (−63.9 + 23.4i)35-s + 32.5i·37-s − 5.41i·41-s + 15.7i·43-s − 32.8·47-s + ⋯ |
| L(s) = 1 | + (0.344 + 0.938i)5-s + 1.94i·7-s − 0.145i·11-s − 1.32i·13-s − 1.24·17-s − 0.615·19-s + 1.06·23-s + (−0.763 + 0.646i)25-s + 1.39i·29-s + 1.24·31-s + (−1.82 + 0.669i)35-s + 0.878i·37-s − 0.132i·41-s + 0.366i·43-s − 0.699·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8970127424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8970127424\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.72 - 4.69i)T \) |
| good | 7 | \( 1 - 13.6iT - 49T^{2} \) |
| 11 | \( 1 + 1.59iT - 121T^{2} \) |
| 13 | \( 1 + 17.2iT - 169T^{2} \) |
| 17 | \( 1 + 21.0T + 289T^{2} \) |
| 19 | \( 1 + 11.6T + 361T^{2} \) |
| 23 | \( 1 - 24.4T + 529T^{2} \) |
| 29 | \( 1 - 40.5iT - 841T^{2} \) |
| 31 | \( 1 - 38.6T + 961T^{2} \) |
| 37 | \( 1 - 32.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.41iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 97.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 88.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 9.07T + 3.72e3T^{2} \) |
| 67 | \( 1 - 26.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 18.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 23.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 147. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100. iT - 9.40e3T^{2} \) |
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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
−9.188855053650386917443128429552, −8.637464845452145050076050810591, −7.920278646650604340860208182277, −6.73782930433166524889037215054, −6.23885404797119453082454120520, −5.45927499812976104113171371489, −4.71886769812606290526577530013, −2.99213926233855199102116318820, −2.88129624976147687425269647416, −1.72124426408465343531404646503,
0.21771617033843884307215372333, 1.23613496872309019393755888201, 2.23898273341210482037387059605, 3.83816392420179023034218063001, 4.41624612295896372386217563813, 4.91676880651828804844616861613, 6.46404905683351356913554260992, 6.69621638300269335875447824162, 7.74588954598707374575834267513, 8.430950185498737223992945286378
