| L(s) = 1 | + (1.13 + 1.48i)2-s + (−0.796 + 1.53i)3-s + (−0.386 + 1.44i)4-s + (−1.43 − 0.0939i)5-s + (−3.18 + 0.568i)6-s + (0.180 + 2.74i)7-s + (0.874 − 0.362i)8-s + (−1.73 − 2.45i)9-s + (−1.49 − 2.23i)10-s + (0.0627 + 0.0550i)11-s + (−1.91 − 1.74i)12-s + (2.54 + 0.682i)13-s + (−3.86 + 3.39i)14-s + (1.28 − 2.13i)15-s + (4.11 + 2.37i)16-s + (0.578 − 4.08i)17-s + ⋯ |
| L(s) = 1 | + (0.804 + 1.04i)2-s + (−0.460 + 0.887i)3-s + (−0.193 + 0.721i)4-s + (−0.641 − 0.0420i)5-s + (−1.30 + 0.232i)6-s + (0.0680 + 1.03i)7-s + (0.309 − 0.128i)8-s + (−0.576 − 0.816i)9-s + (−0.471 − 0.706i)10-s + (0.0189 + 0.0165i)11-s + (−0.551 − 0.503i)12-s + (0.706 + 0.189i)13-s + (−1.03 + 0.906i)14-s + (0.332 − 0.550i)15-s + (1.02 + 0.594i)16-s + (0.140 − 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.578147 + 1.24448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.578147 + 1.24448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.796 - 1.53i)T \) |
| 17 | \( 1 + (-0.578 + 4.08i)T \) |
| good | 2 | \( 1 + (-1.13 - 1.48i)T + (-0.517 + 1.93i)T^{2} \) |
| 5 | \( 1 + (1.43 + 0.0939i)T + (4.95 + 0.652i)T^{2} \) |
| 7 | \( 1 + (-0.180 - 2.74i)T + (-6.94 + 0.913i)T^{2} \) |
| 11 | \( 1 + (-0.0627 - 0.0550i)T + (1.43 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.54 - 0.682i)T + (11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-3.77 - 1.56i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (1.24 + 3.66i)T + (-18.2 + 14.0i)T^{2} \) |
| 29 | \( 1 + (0.471 - 0.232i)T + (17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (-3.80 - 4.33i)T + (-4.04 + 30.7i)T^{2} \) |
| 37 | \( 1 + (0.999 + 5.02i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (2.67 - 5.41i)T + (-24.9 - 32.5i)T^{2} \) |
| 43 | \( 1 + (-1.30 + 9.90i)T + (-41.5 - 11.1i)T^{2} \) |
| 47 | \( 1 + (10.6 - 2.85i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.44 + 10.7i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (2.72 + 2.08i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (11.3 - 0.743i)T + (60.4 - 7.96i)T^{2} \) |
| 67 | \( 1 + (4.02 - 2.32i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.02 + 0.203i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (5.86 + 3.91i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-5.53 + 6.30i)T + (-10.3 - 78.3i)T^{2} \) |
| 83 | \( 1 + (-7.18 + 5.51i)T + (21.4 - 80.1i)T^{2} \) |
| 89 | \( 1 + (9.55 - 9.55i)T - 89iT^{2} \) |
| 97 | \( 1 + (-5.41 - 10.9i)T + (-59.0 + 76.9i)T^{2} \) |
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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
−13.66245814114808473986181533326, −12.22377754329864708520906736396, −11.61726130682160163114120163077, −10.34931006674713385037625765981, −9.107090326638972032865640080179, −7.992458945042824067330302350858, −6.59364884378588759145022878078, −5.59869082142535648474080215298, −4.74457695823053902987227287887, −3.49598892216211996937110746513,
1.38099695745559702355975769635, 3.30094963540546242478247961025, 4.44228010652392603975652950306, 5.91021102456068592730228094790, 7.38545483416880731374189182666, 8.098545761684943291147803629638, 10.12695381446438209832303626879, 11.10282333732563032451449235535, 11.63157095585015577896596699920, 12.55350075594399079277875048121
