L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)4-s + (−0.541 + 0.541i)5-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)8-s + 1.00i·9-s + (0.707 + 0.292i)10-s + (−1.30 + 1.30i)11-s + 12-s + (0.707 + 0.707i)13-s + 0.765·15-s − i·16-s + (0.923 − 0.382i)18-s − 0.765i·20-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)2-s + (−0.707 − 0.707i)3-s + (−0.707 + 0.707i)4-s + (−0.541 + 0.541i)5-s + (−0.382 + 0.923i)6-s + (0.923 + 0.382i)8-s + 1.00i·9-s + (0.707 + 0.292i)10-s + (−1.30 + 1.30i)11-s + 12-s + (0.707 + 0.707i)13-s + 0.765·15-s − i·16-s + (0.923 − 0.382i)18-s − 0.765i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3885552101\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3885552101\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.382 + 0.923i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 5 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 0.765iT - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - 1.84T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 61 | \( 1 + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.84iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (-0.541 - 0.541i)T + iT^{2} \) |
| 89 | \( 1 - 1.84iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.91248546281821422362868763912, −10.34555224713936121295676255561, −9.331087558025482653565714219753, −8.053567689489737256518928589955, −7.53189481950175268137597954106, −6.63351974310866636172805478744, −5.20494614819500860783729876604, −4.27922511120601303739164792078, −2.85007558694573951370941669280, −1.70273815481864581087862194144,
0.56799868889114961379323554390, 3.42137822706251170721136390781, 4.54734357132608278599234076129, 5.51025140663903375047104752966, 5.97212492114191607341116343160, 7.27220233756671052187405079708, 8.398239953341882053270811717965, 8.663351227624629711193134610752, 9.970534337350792000519509291224, 10.62996267143724590448146306485