L(s) = 1 | + (0.587 − 0.809i)3-s + (0.309 + 0.951i)4-s + (−0.587 − 0.809i)5-s + (0.951 − 0.309i)7-s + (−0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s + (0.951 + 0.309i)12-s + (0.363 − 0.5i)13-s − 15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.690i)17-s + (0.587 − 0.809i)20-s + (0.309 − 0.951i)21-s + (−0.309 + 0.951i)25-s + (−0.951 − 0.309i)27-s + (0.587 + 0.809i)28-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s + (0.309 + 0.951i)4-s + (−0.587 − 0.809i)5-s + (0.951 − 0.309i)7-s + (−0.309 − 0.951i)9-s + (0.809 + 0.587i)11-s + (0.951 + 0.309i)12-s + (0.363 − 0.5i)13-s − 15-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.690i)17-s + (0.587 − 0.809i)20-s + (0.309 − 0.951i)21-s + (−0.309 + 0.951i)25-s + (−0.951 − 0.309i)27-s + (0.587 + 0.809i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.351839404\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.351839404\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (0.587 + 0.809i)T \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
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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
−9.619396095321440012337056063446, −8.560155831602474883467279753276, −8.392456097930673210147964565941, −7.51523975599997684846337069339, −6.96362923989851422402673528909, −5.80343910909635239756203287768, −4.22158401684550213104128494473, −3.97592554379337370475311465482, −2.47181757862547321923055113305, −1.39869966022524746553802161121,
1.78859598453094338331194961447, 2.87929122015353144949239620790, 4.01545344951715982630391463511, 4.82413198809587286950998154874, 5.80703762724498235010357044791, 6.80573075021375080997236025567, 7.58885876834610552407467124501, 8.863210100511942331641697017544, 9.006673586540645183268241739612, 10.23631749920631308117999547440