L(s) = 1 | + (−2.75 + 1.18i)3-s − 3.68·5-s + 5.43·7-s + (6.21 − 6.51i)9-s − 1.16·11-s − 14.4i·13-s + (10.1 − 4.35i)15-s − 1.71i·17-s + 28.8i·19-s + (−14.9 + 6.42i)21-s + 35.4i·23-s − 11.4·25-s + (−9.43 + 25.2i)27-s + 33.6·29-s + 55.5·31-s + ⋯ |
L(s) = 1 | + (−0.919 + 0.393i)3-s − 0.736·5-s + 0.776·7-s + (0.690 − 0.723i)9-s − 0.105·11-s − 1.10i·13-s + (0.677 − 0.290i)15-s − 0.100i·17-s + 1.51i·19-s + (−0.714 + 0.305i)21-s + 1.54i·23-s − 0.456·25-s + (−0.349 + 0.936i)27-s + 1.16·29-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.837865 + 0.567065i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.837865 + 0.567065i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.75 - 1.18i)T \) |
good | 5 | \( 1 + 3.68T + 25T^{2} \) |
| 7 | \( 1 - 5.43T + 49T^{2} \) |
| 11 | \( 1 + 1.16T + 121T^{2} \) |
| 13 | \( 1 + 14.4iT - 169T^{2} \) |
| 17 | \( 1 + 1.71iT - 289T^{2} \) |
| 19 | \( 1 - 28.8iT - 361T^{2} \) |
| 23 | \( 1 - 35.4iT - 529T^{2} \) |
| 29 | \( 1 - 33.6T + 841T^{2} \) |
| 31 | \( 1 - 55.5T + 961T^{2} \) |
| 37 | \( 1 - 30.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 63.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 61.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 56.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 49.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 4.73iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 7.08iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 96.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 68.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 9.72T + 6.24e3T^{2} \) |
| 83 | \( 1 - 38.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 0.675iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 86.1T + 9.40e3T^{2} \) |
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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
−11.60175407207486593375654422141, −10.35559078224675977351957462420, −9.852065265440152081560368535278, −8.226369814384602222724041426017, −7.75852197164604372508225293995, −6.37650039321578909323561607975, −5.38668944911650480458504630082, −4.48201607378176314562107557056, −3.34605824989184909877801957159, −1.15413403952308972479848764188,
0.61829304095122317236775229696, 2.28850714727856581605985229117, 4.31646764183003457234512881129, 4.86394310156282979361975467274, 6.30774330932002713467616251149, 7.08055565965471839157549162329, 8.042400511287563774393105979989, 8.972551797919170846014834098319, 10.39825699014697183561464384227, 11.08717348940724422728110493643