L(s) = 1 | − 2.45·2-s + (0.877 + 1.49i)3-s + 4.00·4-s − i·5-s + (−2.15 − 3.66i)6-s − i·7-s − 4.91·8-s + (−1.46 + 2.62i)9-s + 2.45i·10-s + (−0.828 + 3.21i)11-s + (3.51 + 5.98i)12-s + 6.76i·13-s + 2.45i·14-s + (1.49 − 0.877i)15-s + 4.03·16-s − 5.22·17-s + ⋯ |
L(s) = 1 | − 1.73·2-s + (0.506 + 0.862i)3-s + 2.00·4-s − 0.447i·5-s + (−0.877 − 1.49i)6-s − 0.377i·7-s − 1.73·8-s + (−0.486 + 0.873i)9-s + 0.775i·10-s + (−0.249 + 0.968i)11-s + (1.01 + 1.72i)12-s + 1.87i·13-s + 0.655i·14-s + (0.385 − 0.226i)15-s + 1.00·16-s − 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2209564939\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2209564939\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.877 - 1.49i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.828 - 3.21i)T \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 13 | \( 1 - 6.76iT - 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 6.21iT - 19T^{2} \) |
| 23 | \( 1 + 3.22iT - 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 - 2.16T + 31T^{2} \) |
| 37 | \( 1 + 8.03T + 37T^{2} \) |
| 41 | \( 1 + 5.40T + 41T^{2} \) |
| 43 | \( 1 - 6.96iT - 43T^{2} \) |
| 47 | \( 1 + 7.86iT - 47T^{2} \) |
| 53 | \( 1 - 7.52iT - 53T^{2} \) |
| 59 | \( 1 + 4.90iT - 59T^{2} \) |
| 61 | \( 1 + 5.13iT - 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 8.09iT - 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 4.27T + 83T^{2} \) |
| 89 | \( 1 - 7.30iT - 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.932872676322318470482236607648, −9.323362676316731277094962820221, −8.815488174145695730550236132119, −8.215228067756004190542889918760, −7.00795401681582209676922482028, −6.70556342504692929325025936984, −4.80471812920888597656493286805, −4.30998612928058203979020725236, −2.59625698069730211128468878261, −1.76245234389826889079843727244,
0.15359171180717028852848196355, 1.48933406857691138882992676644, 2.63102311333875128984309595312, 3.37230088243511787830608090751, 5.61829144779809627615361329295, 6.32338953691971175343810649230, 7.22539706789809062784661924042, 8.000469445957596167245664179274, 8.421895441658861738934797262387, 9.067305177682494556880946333127