L(s) = 1 | + (−0.494 − 1.32i)2-s + (0.862 + 1.49i)3-s + (−1.51 + 1.31i)4-s + (−3.22 + 1.86i)5-s + (1.55 − 1.88i)6-s + (−0.0286 + 0.0496i)7-s + (2.48 + 1.35i)8-s + (0.0138 − 0.0240i)9-s + (4.06 + 3.35i)10-s + 6.50i·11-s + (−3.25 − 1.12i)12-s + (−1.17 + 2.04i)13-s + (0.0800 + 0.0134i)14-s + (−5.56 − 3.21i)15-s + (0.566 − 3.95i)16-s + (1.70 − 2.95i)17-s + ⋯ |
L(s) = 1 | + (−0.349 − 0.936i)2-s + (0.497 + 0.862i)3-s + (−0.755 + 0.655i)4-s + (−1.44 + 0.832i)5-s + (0.633 − 0.767i)6-s + (−0.0108 + 0.0187i)7-s + (0.877 + 0.478i)8-s + (0.00463 − 0.00802i)9-s + (1.28 + 1.06i)10-s + 1.96i·11-s + (−0.940 − 0.325i)12-s + (−0.326 + 0.565i)13-s + (0.0213 + 0.00359i)14-s + (−1.43 − 0.828i)15-s + (0.141 − 0.989i)16-s + (0.413 − 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.624963 + 0.428234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.624963 + 0.428234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.494 + 1.32i)T \) |
| 43 | \( 1 + (2.55 + 6.03i)T \) |
good | 3 | \( 1 + (-0.862 - 1.49i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (3.22 - 1.86i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.0286 - 0.0496i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 6.50iT - 11T^{2} \) |
| 13 | \( 1 + (1.17 - 2.04i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 2.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.342 - 0.594i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.84 - 1.64i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.67 + 2.12i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.93 + 2.27i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.37 + 4.25i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 47 | \( 1 - 8.53iT - 47T^{2} \) |
| 53 | \( 1 + (-2.84 - 4.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 7.15iT - 59T^{2} \) |
| 61 | \( 1 + (3.73 + 2.15i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.47 - 2.58i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.38 + 2.39i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.1 - 5.83i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.2 - 5.94i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.33 + 3.65i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.92 - 3.41i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8.61T + 97T^{2} \) |
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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
−12.40346336557379767518303268194, −11.92950598510729485387147460554, −10.89664989995222811641152216100, −9.867724159569029478717132988211, −9.342365050435105878916095072144, −7.83288832011001396506113153766, −7.20067080327293093308027661971, −4.48678539098109781508435094553, −3.96188236479768422116101026052, −2.62567491691325460033864237275,
0.796484891447544447858617490681, 3.60966290454972247929397170730, 5.08695796548938626850231779340, 6.43088091140708622859974107238, 7.87429632137698592855576517189, 8.064984870169292256669123278969, 8.873284657737268737918879821007, 10.51520357881039751368263071669, 11.71736321425861718015136828561, 12.86050141391167360304420486788