L(s) = 1 | + (−0.628 − 2.93i)3-s − 6.51·5-s − 7.64·7-s + (−8.21 + 3.68i)9-s + 17.8·11-s − 14.4i·13-s + (4.09 + 19.1i)15-s + 18.6i·17-s + 12.9i·19-s + (4.80 + 22.4i)21-s + 28.1i·23-s + 17.4·25-s + (15.9 + 21.7i)27-s − 3.08·29-s + 32.1·31-s + ⋯ |
L(s) = 1 | + (−0.209 − 0.977i)3-s − 1.30·5-s − 1.09·7-s + (−0.912 + 0.409i)9-s + 1.62·11-s − 1.10i·13-s + (0.272 + 1.27i)15-s + 1.09i·17-s + 0.682i·19-s + (0.228 + 1.06i)21-s + 1.22i·23-s + 0.696·25-s + (0.591 + 0.806i)27-s − 0.106·29-s + 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.533040 + 0.289925i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.533040 + 0.289925i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.628 + 2.93i)T \) |
good | 5 | \( 1 + 6.51T + 25T^{2} \) |
| 7 | \( 1 + 7.64T + 49T^{2} \) |
| 11 | \( 1 - 17.8T + 121T^{2} \) |
| 13 | \( 1 + 14.4iT - 169T^{2} \) |
| 17 | \( 1 - 18.6iT - 289T^{2} \) |
| 19 | \( 1 - 12.9iT - 361T^{2} \) |
| 23 | \( 1 - 28.1iT - 529T^{2} \) |
| 29 | \( 1 + 3.08T + 841T^{2} \) |
| 31 | \( 1 - 32.1T + 961T^{2} \) |
| 37 | \( 1 + 1.57iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 86.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 11.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 91.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 17.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 58.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 123.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 81.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 142. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58.1T + 9.40e3T^{2} \) |
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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
−11.55947099223963117614761980460, −10.52980929961655578855074346741, −9.308845069824950993521355494358, −8.243348324144213324801174011157, −7.54810608678994510844416805607, −6.55096437083710620223140912178, −5.81380876102269501627709585715, −4.00476267484221536847139522210, −3.18559570736926227462955638891, −1.18524683722328429488665646965,
0.32329091844723310516791895937, 3.05407563610276593959809720139, 4.01964952695429945050294847036, 4.67643500308750799138897023422, 6.40455136845121215018811114454, 6.93626354719758498953167365193, 8.487944358006434609179767117674, 9.243722717029624720616221944968, 9.900531013263916818287962836853, 11.23698052292841620044348720078