L(s) = 1 | + (0.5 − 0.866i)5-s + (1.5 + 2.59i)7-s + (−1.5 − 2.59i)11-s + 4·17-s − 6·19-s + (3 − 5.19i)23-s + (2 + 3.46i)25-s + (1 + 1.73i)29-s + (4.5 − 7.79i)31-s + 3·35-s − 2·37-s + (5 − 8.66i)41-s + (3 + 5.19i)43-s + (3 + 5.19i)47-s + (−1 + 1.73i)49-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (0.566 + 0.981i)7-s + (−0.452 − 0.783i)11-s + 0.970·17-s − 1.37·19-s + (0.625 − 1.08i)23-s + (0.400 + 0.692i)25-s + (0.185 + 0.321i)29-s + (0.808 − 1.39i)31-s + 0.507·35-s − 0.328·37-s + (0.780 − 1.35i)41-s + (0.457 + 0.792i)43-s + (0.437 + 0.757i)47-s + (−0.142 + 0.247i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.990665785\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.990665785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.5 + 7.79i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 13T + 53T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3 + 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.819346814604100255739101658364, −8.226120024367432596341438664645, −7.48196711426851802790243109007, −6.30550073494267148257715964716, −5.72556258867310486551437243281, −5.02501958303890848718776280880, −4.15336891468425782759188445146, −2.90752481199080668703383881696, −2.16219155677443048109132050131, −0.820894858939716181770140725296,
1.02292980729207500018704471317, 2.15595654771543905306353677548, 3.20138194646257024700427056504, 4.24956306862705455162588548021, 4.87361384513441165614587573010, 5.83947962666259306511394282681, 6.81258704366845437877083106946, 7.34236234737830867441348407904, 8.075839801156415310934025897735, 8.841060950394308479581827706312