L(s) = 1 | + (−0.739 + 0.673i)2-s + (0.0922 − 0.995i)4-s + (−1.27 − 0.961i)5-s + (0.602 + 0.798i)8-s + (0.850 + 0.526i)9-s + (1.58 − 0.147i)10-s + (−1.02 − 1.35i)13-s + (−0.982 − 0.183i)16-s + (−0.982 − 0.183i)17-s + (−0.982 + 0.183i)18-s + (−1.07 + 1.17i)20-s + (0.423 + 1.48i)25-s + (1.67 + 0.312i)26-s + (−1.04 − 0.0971i)29-s + (0.850 − 0.526i)32-s + ⋯ |
L(s) = 1 | + (−0.739 + 0.673i)2-s + (0.0922 − 0.995i)4-s + (−1.27 − 0.961i)5-s + (0.602 + 0.798i)8-s + (0.850 + 0.526i)9-s + (1.58 − 0.147i)10-s + (−1.02 − 1.35i)13-s + (−0.982 − 0.183i)16-s + (−0.982 − 0.183i)17-s + (−0.982 + 0.183i)18-s + (−1.07 + 1.17i)20-s + (0.423 + 1.48i)25-s + (1.67 + 0.312i)26-s + (−1.04 − 0.0971i)29-s + (0.850 − 0.526i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3265661118\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3265661118\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.739 - 0.673i)T \) |
| 17 | \( 1 + (0.982 + 0.183i)T \) |
good | 3 | \( 1 + (-0.850 - 0.526i)T^{2} \) |
| 5 | \( 1 + (1.27 + 0.961i)T + (0.273 + 0.961i)T^{2} \) |
| 7 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 11 | \( 1 + (-0.982 - 0.183i)T^{2} \) |
| 13 | \( 1 + (1.02 + 1.35i)T + (-0.273 + 0.961i)T^{2} \) |
| 19 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 23 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 29 | \( 1 + (1.04 + 0.0971i)T + (0.982 + 0.183i)T^{2} \) |
| 31 | \( 1 + (-0.273 + 0.961i)T^{2} \) |
| 37 | \( 1 + (0.486 + 1.25i)T + (-0.739 + 0.673i)T^{2} \) |
| 41 | \( 1 + (1.53 + 0.436i)T + (0.850 + 0.526i)T^{2} \) |
| 43 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 47 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 53 | \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \) |
| 59 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 61 | \( 1 + (-1.72 + 0.857i)T + (0.602 - 0.798i)T^{2} \) |
| 67 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 71 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 73 | \( 1 + (-0.193 + 1.03i)T + (-0.932 - 0.361i)T^{2} \) |
| 79 | \( 1 + (0.0922 + 0.995i)T^{2} \) |
| 83 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 89 | \( 1 + (-0.890 + 1.17i)T + (-0.273 - 0.961i)T^{2} \) |
| 97 | \( 1 + (1.04 - 1.69i)T + (-0.445 - 0.895i)T^{2} \) |
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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
−9.607305750895616519938695028713, −8.759341784135998873997691933779, −7.996345463500919967373758999904, −7.53204577814291255973715747278, −6.77802117867212991139396934752, −5.26320703712977998300950031543, −4.90527774524980488591202342043, −3.78727712710552147652770600707, −2.01101349520912127598580525766, −0.36238845056206168124369861738,
1.82648247560790348177308102230, 3.03711286703193514377961882709, 3.97800891149683069888548978462, 4.57112949563601441202357055163, 6.80187249312404601932640324185, 6.87006980199327037082034129025, 7.77260556351829262159940876608, 8.669203121020271736110216016612, 9.558206645866012204680972232361, 10.18136259815685029684262720580