L(s) = 1 | + (1.84 + 1.25i)2-s + (1.09 + 2.78i)4-s + (1.49 − 0.460i)5-s + (1.29 + 2.30i)7-s + (−0.495 + 2.17i)8-s + (3.33 + 1.02i)10-s + (0.278 − 3.71i)11-s + (−0.768 + 0.370i)13-s + (−0.524 + 5.89i)14-s + (0.742 − 0.689i)16-s + (−6.99 + 1.05i)17-s + (0.365 + 0.633i)19-s + (2.91 + 3.65i)20-s + (5.19 − 6.50i)22-s + (−1.11 − 0.168i)23-s + ⋯ |
L(s) = 1 | + (1.30 + 0.890i)2-s + (0.547 + 1.39i)4-s + (0.667 − 0.205i)5-s + (0.487 + 0.872i)7-s + (−0.175 + 0.767i)8-s + (1.05 + 0.325i)10-s + (0.0839 − 1.11i)11-s + (−0.213 + 0.102i)13-s + (−0.140 + 1.57i)14-s + (0.185 − 0.172i)16-s + (−1.69 + 0.255i)17-s + (0.0839 + 0.145i)19-s + (0.652 + 0.818i)20-s + (1.10 − 1.38i)22-s + (−0.233 − 0.0351i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.44667 + 1.77320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.44667 + 1.77320i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.29 - 2.30i)T \) |
good | 2 | \( 1 + (-1.84 - 1.25i)T + (0.730 + 1.86i)T^{2} \) |
| 5 | \( 1 + (-1.49 + 0.460i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.278 + 3.71i)T + (-10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (0.768 - 0.370i)T + (8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (6.99 - 1.05i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.365 - 0.633i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.11 + 0.168i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-2.81 - 3.53i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (3.36 - 5.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.19 + 8.14i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-2.11 + 9.27i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (1.48 + 6.52i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (2.66 + 1.81i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.22 - 5.65i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (4.93 + 1.52i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-2.49 + 6.36i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-7.43 + 12.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.24 - 7.83i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (3.44 - 2.34i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (1.41 + 2.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (13.8 + 6.64i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.515 - 6.88i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 - 7.03T + 97T^{2} \) |
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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
−11.55212372228142707000291477987, −10.63113708378729072721155620841, −9.139650506038351609642032135568, −8.552373672010428252674030926379, −7.29483855875109890412049178721, −6.25569403423277811628647252076, −5.62561187960135650698790389371, −4.83103548281003009609490508947, −3.61673489067542602720947017384, −2.18810547907624731672698846944,
1.75002891683150883821716452784, 2.70874801827380841405554834547, 4.33168103816650415010451369473, 4.61892774372509181598210581464, 5.99687656323088973525522402948, 6.92432681867058827486619751530, 8.119497839867770574983787024458, 9.690928212937473050534720638465, 10.19535153142692028011312948221, 11.28836712942498625322875378230