| L(s) = 1 | + (−1.33 + 0.459i)2-s + (1.70 − 0.304i)3-s + (1.57 − 1.22i)4-s + (−2.29 + 1.04i)5-s + (−2.14 + 1.19i)6-s + (−0.382 − 0.401i)7-s + (−1.54 + 2.36i)8-s + (2.81 − 1.03i)9-s + (2.58 − 2.45i)10-s + (3.69 + 0.352i)11-s + (2.31 − 2.57i)12-s + (−1.78 − 0.343i)13-s + (0.696 + 0.361i)14-s + (−3.58 + 2.48i)15-s + (0.980 − 3.87i)16-s + (1.28 − 2.49i)17-s + ⋯ |
| L(s) = 1 | + (−0.945 + 0.324i)2-s + (0.984 − 0.175i)3-s + (0.789 − 0.614i)4-s + (−1.02 + 0.467i)5-s + (−0.874 + 0.485i)6-s + (−0.144 − 0.151i)7-s + (−0.546 + 0.837i)8-s + (0.938 − 0.345i)9-s + (0.817 − 0.775i)10-s + (1.11 + 0.106i)11-s + (0.668 − 0.743i)12-s + (−0.494 − 0.0953i)13-s + (0.186 + 0.0965i)14-s + (−0.926 + 0.640i)15-s + (0.245 − 0.969i)16-s + (0.312 − 0.605i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27555 + 0.0997449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27555 + 0.0997449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.459i)T \) |
| 3 | \( 1 + (-1.70 + 0.304i)T \) |
| 67 | \( 1 + (7.02 + 4.20i)T \) |
| good | 5 | \( 1 + (2.29 - 1.04i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (0.382 + 0.401i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-3.69 - 0.352i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (1.78 + 0.343i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (-1.28 + 2.49i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 1.71i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-3.38 + 1.35i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (-1.80 - 1.04i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.19 - 6.21i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (-2.53 - 4.39i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.71 + 0.462i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (-1.51 - 2.35i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-2.11 - 1.66i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (-1.46 + 2.27i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (5.90 + 6.81i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-12.4 + 1.18i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (10.5 - 5.41i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-1.20 + 0.115i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (-9.81 + 3.39i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (4.24 - 5.96i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (-10.4 + 1.50i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.46 - 4.26i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.988979251677342684919570638195, −9.297478402001897239639469174743, −8.588158629900474684794950984589, −7.66486606188746160829870876645, −7.14742299204987157628736070090, −6.48103226468870568813166603701, −4.81110196326887277940129168610, −3.54135528941443230813274046662, −2.64555926170698401319893581358, −1.04761903364842106736517418472,
1.11043393406490048736170377443, 2.53723855802592955917419273722, 3.70270757591059389809563148195, 4.29902678502277258237073931108, 6.10245688718580034137417016968, 7.38596680916155602515895413378, 7.75086435617249157852497286629, 8.725909927382030189858542941881, 9.214316462297723378673605490249, 9.977242695538107286962768567185
