L(s) = 1 | + (−0.334 − 0.334i)5-s + 4.55i·7-s + (−2.47 − 2.47i)11-s + (0.0594 − 0.0594i)13-s − 3.61·17-s + (−2.55 + 2.55i)19-s + 2.82i·23-s − 4.77i·25-s + (−5.16 + 5.16i)29-s − 0.557·31-s + (1.52 − 1.52i)35-s + (−4.38 − 4.38i)37-s − 9.27i·41-s + (1.61 + 1.61i)43-s − 2.82·47-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.149i)5-s + 1.72i·7-s + (−0.745 − 0.745i)11-s + (0.0164 − 0.0164i)13-s − 0.877·17-s + (−0.586 + 0.586i)19-s + 0.589i·23-s − 0.955i·25-s + (−0.958 + 0.958i)29-s − 0.100·31-s + (0.258 − 0.258i)35-s + (−0.721 − 0.721i)37-s − 1.44i·41-s + (0.245 + 0.245i)43-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4581911828\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4581911828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.334 + 0.334i)T + 5iT^{2} \) |
| 7 | \( 1 - 4.55iT - 7T^{2} \) |
| 11 | \( 1 + (2.47 + 2.47i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.0594 + 0.0594i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + (2.55 - 2.55i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (5.16 - 5.16i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.557T + 31T^{2} \) |
| 37 | \( 1 + (4.38 + 4.38i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.27iT - 41T^{2} \) |
| 43 | \( 1 + (-1.61 - 1.61i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (0.493 + 0.493i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4 - 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.72 - 2.72i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.77 - 3.77i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.11iT - 71T^{2} \) |
| 73 | \( 1 - 0.541iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + (10.6 - 10.6i)T - 83iT^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−10.19631657810327981073283521840, −9.016285176758913462165746234666, −8.726264590431806147497820042359, −7.908544136608415736844894939363, −6.76956026171077010708621248393, −5.70290480613880822249814185573, −5.37725670525958057663841771758, −4.04543000785197461797650593148, −2.84938032313558456550956146574, −1.97168152047575792384849187507,
0.18471419669255366963919380793, 1.83001800410577286778430828608, 3.19501121344931860843119672528, 4.30337041020132721349178179599, 4.80967045397779155351749915376, 6.24523366405332067897016669133, 7.11273504294284353714193270781, 7.56349501738674329348473432181, 8.517352120409288279894168086258, 9.619082451869839691504932724277