| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.820231 - 1.21193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.820231 - 1.21193i\) |
| \(L(2)\) |
|
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
−10.77545460802300538563802455265, −9.803337729183953187450158329193, −8.862021653837305996078121659277, −8.087147155627495046590126235052, −7.17083689180790210658366387947, −6.32880556234548197648519840810, −4.69440258925883297709367545773, −3.47051286975971968140074676250, −2.61381084829120786843213132376, −0.57111821526368435928969081483,
1.92972045022045562489897659419, 3.57720466451577907837346264548, 3.96902245123951687544426862897, 5.53849078140131770868601741077, 6.94732708973697589602234523778, 7.72454013737923171148271944294, 8.727421025261125989171760838612, 9.520717863378735678283732283679, 10.28052582873209514083388286612, 11.43449136400417826947178546956
