L(s) = 1 | + (−1.5 + 0.866i)3-s + 2·4-s + (2 − 1.73i)7-s + (1.5 − 2.59i)9-s + (−3 + 1.73i)12-s + (−1 + 3.46i)13-s + 4·16-s + (7.5 + 4.33i)19-s + (−1.50 + 4.33i)21-s + (2.5 − 4.33i)25-s + 5.19i·27-s + (4 − 3.46i)28-s + (−7.5 − 4.33i)31-s + (3 − 5.19i)36-s − 11·37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + 4-s + (0.755 − 0.654i)7-s + (0.5 − 0.866i)9-s + (−0.866 + 0.499i)12-s + (−0.277 + 0.960i)13-s + 16-s + (1.72 + 0.993i)19-s + (−0.327 + 0.944i)21-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (0.755 − 0.654i)28-s + (−1.34 − 0.777i)31-s + (0.5 − 0.866i)36-s − 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30463 + 0.191328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30463 + 0.191328i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 0.866i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 - 2T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-7.5 - 4.33i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + (13.5 + 7.79i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (12 + 6.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.80201489812834484710536665064, −11.04352919388191984109753968112, −10.34030112497915912392582765464, −9.358892255116999374650434056434, −7.75752812144092765319546872157, −6.99803199730055645006653172379, −5.91618307430411832413316018983, −4.83483730757439600057775632918, −3.56940680954281740674240294168, −1.55479489252007004726710845250,
1.49903082663164853734163490938, 2.96282131225023266566178098366, 5.18736756205102008593806676804, 5.64101106934306220767374995501, 7.11526932273722673105108455125, 7.53654773184354997369529670453, 8.910111944363342481758103982556, 10.39319952387523364057149836898, 11.05492240262172909463277311833, 11.87147962314324408927532650214