L(s) = 1 | + (−0.903 + 1.08i)2-s + (−0.368 − 1.96i)4-s + (2.09 + 0.770i)5-s + 3.05·7-s + (2.47 + 1.37i)8-s + (−2.73 + 1.58i)10-s + (1.80 − 1.80i)11-s + (2.47 − 2.47i)13-s + (−2.75 + 3.31i)14-s + (−3.72 + 1.44i)16-s − 3.66i·17-s + (−2.31 − 2.31i)19-s + (0.741 − 4.41i)20-s + (0.334 + 3.60i)22-s − 4.86·23-s + ⋯ |
L(s) = 1 | + (−0.638 + 0.769i)2-s + (−0.184 − 0.982i)4-s + (0.938 + 0.344i)5-s + 1.15·7-s + (0.873 + 0.486i)8-s + (−0.864 + 0.502i)10-s + (0.545 − 0.545i)11-s + (0.685 − 0.685i)13-s + (−0.736 + 0.887i)14-s + (−0.932 + 0.361i)16-s − 0.889i·17-s + (−0.531 − 0.531i)19-s + (0.165 − 0.986i)20-s + (0.0713 + 0.768i)22-s − 1.01·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45958 + 0.304539i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45958 + 0.304539i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.903 - 1.08i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.09 - 0.770i)T \) |
good | 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + (-1.80 + 1.80i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.47 + 2.47i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.66iT - 17T^{2} \) |
| 19 | \( 1 + (2.31 + 2.31i)T + 19iT^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + (4.74 + 4.74i)T + 29iT^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + (-5.40 - 5.40i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.47iT - 41T^{2} \) |
| 43 | \( 1 + (4.19 + 4.19i)T + 43iT^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 + (-9.99 - 9.99i)T + 53iT^{2} \) |
| 59 | \( 1 + (-2.47 + 2.47i)T - 59iT^{2} \) |
| 61 | \( 1 + (-8.01 - 8.01i)T + 61iT^{2} \) |
| 67 | \( 1 + (8.60 - 8.60i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.63iT - 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + (3.65 - 3.65i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−10.36746598865618831708460730512, −9.479455385579266166456527089726, −8.680989745979063760130498973540, −7.960148527318181477934379976164, −6.99095750316759589075973381004, −6.01280864025049796019700909247, −5.44088018743843400904590619391, −4.26617788460014428216427785946, −2.41961891670945598454894169780, −1.14632812441912611218284827156,
1.54121270028190164350798899055, 2.00644438240468454518997545125, 3.82127732880201331274900212861, 4.64031164670971847789371939416, 5.90771988245784097554526602024, 6.97248281813378616236060807377, 8.213486850902601953472167059901, 8.634598457116550747515874264404, 9.614368074457860877960940560566, 10.26504243027507787688445022210