L(s) = 1 | + (−0.965 + 0.258i)2-s + (1.71 − 0.266i)3-s + (0.866 − 0.499i)4-s + (−0.828 − 0.828i)5-s + (−1.58 + 0.700i)6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (2.85 − 0.911i)9-s + (1.01 + 0.585i)10-s + (1.25 + 4.67i)11-s + (1.34 − 1.08i)12-s + (0.0707 − 3.60i)13-s + i·14-s + (−1.63 − 1.19i)15-s + (0.500 − 0.866i)16-s + (0.258 + 0.448i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.988 − 0.153i)3-s + (0.433 − 0.249i)4-s + (−0.370 − 0.370i)5-s + (−0.646 + 0.285i)6-s + (0.0978 − 0.365i)7-s + (−0.249 + 0.249i)8-s + (0.952 − 0.303i)9-s + (0.320 + 0.185i)10-s + (0.377 + 1.40i)11-s + (0.389 − 0.313i)12-s + (0.0196 − 0.999i)13-s + 0.267i·14-s + (−0.423 − 0.309i)15-s + (0.125 − 0.216i)16-s + (0.0627 + 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45549 - 0.315916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45549 - 0.315916i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 + (-1.71 + 0.266i)T \) |
| 7 | \( 1 + (-0.258 + 0.965i)T \) |
| 13 | \( 1 + (-0.0707 + 3.60i)T \) |
good | 5 | \( 1 + (0.828 + 0.828i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.25 - 4.67i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.258 - 0.448i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.50 - 1.47i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.54 + 4.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.83 + 1.06i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.11 + 4.11i)T - 31iT^{2} \) |
| 37 | \( 1 + (9.86 - 2.64i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-6.46 + 1.73i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (0.253 - 0.146i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (8.54 - 8.54i)T - 47iT^{2} \) |
| 53 | \( 1 + 9.52iT - 53T^{2} \) |
| 59 | \( 1 + (1.53 + 0.411i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.90 - 10.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.99 - 11.1i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.487 - 1.81i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.68 - 4.68i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 + (11.7 + 11.7i)T + 83iT^{2} \) |
| 89 | \( 1 + (-0.717 - 2.67i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.95 - 1.05i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.24844862279041375079315811037, −9.875827653917338348913548532377, −8.879481878552499312266489963758, −8.027170927816162029063528450322, −7.47042515552971744136543152880, −6.58597153946674296360163577899, −5.01062110525195352880156991134, −3.91840097198890822236010613600, −2.59005571506184228005735438718, −1.16614019255962515743100062280,
1.51001566940130424881945615759, 3.03724185960524981316623821437, 3.64079621861607483495207318387, 5.23599756307169174111775846330, 6.69537900777489833035526727298, 7.45360223400537401654698578391, 8.399956984450844984632623789933, 9.103937564372702390462149734609, 9.659300493412295856011699452029, 10.94076351741669427750773998483