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 | \( 1 + 15.4T + 97T^{2} \) |
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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
−9.067305177682494556880946333127, −8.421895441658861738934797262387, −8.000469445957596167245664179274, −7.22539706789809062784661924042, −6.32338953691971175343810649230, −5.61829144779809627615361329295, −3.37230088243511787830608090751, −2.63102311333875128984309595312, −1.48933406857691138882992676644, −0.15359171180717028852848196355,
1.76245234389826889079843727244, 2.59625698069730211128468878261, 4.30998612928058203979020725236, 4.80471812920888597656493286805, 6.70556342504692929325025936984, 7.00795401681582209676922482028, 8.215228067756004190542889918760, 8.815488174145695730550236132119, 9.323362676316731277094962820221, 9.932872676322318470482236607648