| L(s) = 1 | + (−1.82 − 0.0209i)2-s + (−0.585 + 2.56i)3-s + (1.32 + 0.0305i)4-s + (−1.34 + 2.52i)5-s + (1.12 − 4.66i)6-s + (−0.483 − 1.19i)7-s + (1.22 + 0.0423i)8-s + (−3.52 − 1.69i)9-s + (2.50 − 4.57i)10-s + (−4.19 − 4.37i)11-s + (−0.854 + 3.38i)12-s + (4.43 + 0.669i)13-s + (0.856 + 2.18i)14-s + (−5.67 − 4.91i)15-s + (−4.88 − 0.225i)16-s + (1.03 − 6.15i)17-s + ⋯ |
| L(s) = 1 | + (−1.28 − 0.0148i)2-s + (−0.337 + 1.48i)3-s + (0.663 + 0.0152i)4-s + (−0.599 + 1.12i)5-s + (0.457 − 1.90i)6-s + (−0.182 − 0.450i)7-s + (0.433 + 0.0149i)8-s + (−1.17 − 0.566i)9-s + (0.790 − 1.44i)10-s + (−1.26 − 1.31i)11-s + (−0.246 + 0.977i)12-s + (1.23 + 0.185i)13-s + (0.228 + 0.583i)14-s + (−1.46 − 1.26i)15-s + (−1.22 − 0.0562i)16-s + (0.251 − 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.252876 - 0.0589024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.252876 - 0.0589024i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + (20.5 + 11.0i)T \) |
| good | 2 | \( 1 + (1.82 + 0.0209i)T + (1.99 + 0.0460i)T^{2} \) |
| 3 | \( 1 + (0.585 - 2.56i)T + (-2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (1.34 - 2.52i)T + (-2.79 - 4.14i)T^{2} \) |
| 7 | \( 1 + (0.483 + 1.19i)T + (-5.02 + 4.87i)T^{2} \) |
| 11 | \( 1 + (4.19 + 4.37i)T + (-0.442 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.43 - 0.669i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-1.03 + 6.15i)T + (-16.0 - 5.56i)T^{2} \) |
| 19 | \( 1 + (0.803 - 0.433i)T + (10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (3.89 + 0.913i)T + (20.6 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.04 + 1.16i)T + (14.7 + 24.9i)T^{2} \) |
| 31 | \( 1 + (-6.95 - 2.50i)T + (23.8 + 19.7i)T^{2} \) |
| 37 | \( 1 + (3.02 - 2.44i)T + (7.81 - 36.1i)T^{2} \) |
| 41 | \( 1 + (2.40 + 4.16i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.0164 + 0.167i)T + (-42.1 - 8.35i)T^{2} \) |
| 47 | \( 1 + (0.355 - 0.0286i)T + (46.3 - 7.53i)T^{2} \) |
| 53 | \( 1 + (-3.56 + 1.23i)T + (41.6 - 32.8i)T^{2} \) |
| 59 | \( 1 + (8.91 - 3.79i)T + (40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (1.88 - 0.878i)T + (39.1 - 46.8i)T^{2} \) |
| 67 | \( 1 + (6.72 + 6.53i)T + (1.92 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-14.7 + 2.91i)T + (65.6 - 27.0i)T^{2} \) |
| 73 | \( 1 + (-8.25 + 7.14i)T + (10.4 - 72.2i)T^{2} \) |
| 79 | \( 1 + (6.30 + 4.51i)T + (25.4 + 74.7i)T^{2} \) |
| 83 | \( 1 + (5.22 + 3.30i)T + (35.5 + 74.9i)T^{2} \) |
| 89 | \( 1 + (-0.418 - 4.83i)T + (-87.6 + 15.2i)T^{2} \) |
| 97 | \( 1 + (1.89 + 14.2i)T + (-93.6 + 25.3i)T^{2} \) |
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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
−10.50886043853110998648527579759, −10.15243626169411966166026991570, −9.082132795597632234398938075049, −8.272179118464174839556501232963, −7.43382319027339380438706809644, −6.28718171186019564357651610359, −5.05053491813420902085398158399, −3.82994808516525938639068243034, −2.97570168046591155509032551281, −0.28301499229906103402384244257,
1.14250466714448582965295504380, 2.10091958141834316644887631580, 4.28882101181908579924096052821, 5.58164797024393933995732599206, 6.61408847871128155349833012841, 7.77278372985121768132774507775, 8.074000629262323685336226525854, 8.732188723706370635120494952179, 9.896507361033616381367208552515, 10.76132174886903551771142598155
