L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.46 − 1.12i)5-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (1.46 + 1.12i)10-s + 2i·13-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (0.499 + 0.866i)18-s + (−0.707 + 1.70i)20-s + (0.624 − 2.33i)25-s + (−1.93 + 0.517i)26-s + (−0.707 − 0.292i)29-s + (0.965 + 0.258i)32-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.46 − 1.12i)5-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (1.46 + 1.12i)10-s + 2i·13-s + (0.500 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (0.499 + 0.866i)18-s + (−0.707 + 1.70i)20-s + (0.624 − 2.33i)25-s + (−1.93 + 0.517i)26-s + (−0.707 − 0.292i)29-s + (0.965 + 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.804476851\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804476851\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.258 - 0.965i)T \) |
good | 3 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 1.12i)T + (0.258 - 0.965i)T^{2} \) |
| 11 | \( 1 + (0.258 + 0.965i)T^{2} \) |
| 13 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 29 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 37 | \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.0999 + 0.758i)T + (-0.965 - 0.258i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \) |
| 79 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \) |
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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
−8.902262016985106766447445118057, −8.354025500616468348319208648740, −7.17300320525373877394062280710, −6.64372085525427179602000512477, −5.92385264451569795346653169382, −5.28656099825746593670920915366, −4.29721181045590722963706516135, −4.05072825940444834206908089123, −2.18579476032607385604497716962, −1.36983094788062613065812156125,
1.22432762265950719599851836811, 2.31023400830011018112165371143, 2.87600148813840099622962838192, 3.71169838320893329692353108403, 5.01031281738257404716526739051, 5.46901808378704114361262730222, 6.26762385377529480300486366786, 7.13695737364886439537249098697, 7.920428807678783620556321116878, 9.082033511390151199235309554557