| L(s) = 1 | + (−0.956 − 0.0110i)2-s + (−0.143 + 0.628i)3-s + (−1.08 − 0.0249i)4-s + (−1.82 + 3.43i)5-s + (0.144 − 0.599i)6-s + (0.431 + 1.06i)7-s + (2.94 + 0.101i)8-s + (2.32 + 1.12i)9-s + (1.78 − 3.26i)10-s + (0.710 + 0.739i)11-s + (0.171 − 0.678i)12-s + (0.0967 + 0.0145i)13-s + (−0.400 − 1.02i)14-s + (−1.89 − 1.64i)15-s + (−0.649 − 0.0299i)16-s + (−0.853 + 5.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.676 − 0.00778i)2-s + (−0.0828 + 0.362i)3-s + (−0.542 − 0.0124i)4-s + (−0.817 + 1.53i)5-s + (0.0588 − 0.244i)6-s + (0.163 + 0.401i)7-s + (1.04 + 0.0360i)8-s + (0.776 + 0.373i)9-s + (0.564 − 1.03i)10-s + (0.214 + 0.223i)11-s + (0.0494 − 0.195i)12-s + (0.0268 + 0.00404i)13-s + (−0.107 − 0.272i)14-s + (−0.489 − 0.423i)15-s + (−0.162 − 0.00747i)16-s + (−0.207 + 1.22i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00682328 + 0.529667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00682328 + 0.529667i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + (-23.3 - 1.45i)T \) |
| good | 2 | \( 1 + (0.956 + 0.0110i)T + (1.99 + 0.0460i)T^{2} \) |
| 3 | \( 1 + (0.143 - 0.628i)T + (-2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (1.82 - 3.43i)T + (-2.79 - 4.14i)T^{2} \) |
| 7 | \( 1 + (-0.431 - 1.06i)T + (-5.02 + 4.87i)T^{2} \) |
| 11 | \( 1 + (-0.710 - 0.739i)T + (-0.442 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0967 - 0.0145i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (0.853 - 5.06i)T + (-16.0 - 5.56i)T^{2} \) |
| 19 | \( 1 + (0.515 - 0.277i)T + (10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (2.17 + 0.510i)T + (20.6 + 10.2i)T^{2} \) |
| 29 | \( 1 + (5.77 + 3.29i)T + (14.7 + 24.9i)T^{2} \) |
| 31 | \( 1 + (2.73 + 0.982i)T + (23.8 + 19.7i)T^{2} \) |
| 37 | \( 1 + (2.75 - 2.22i)T + (7.81 - 36.1i)T^{2} \) |
| 41 | \( 1 + (-3.51 - 6.08i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.581 + 5.92i)T + (-42.1 - 8.35i)T^{2} \) |
| 47 | \( 1 + (-10.6 + 0.859i)T + (46.3 - 7.53i)T^{2} \) |
| 53 | \( 1 + (0.0140 - 0.00486i)T + (41.6 - 32.8i)T^{2} \) |
| 59 | \( 1 + (-0.555 + 0.236i)T + (40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (-4.24 + 1.98i)T + (39.1 - 46.8i)T^{2} \) |
| 67 | \( 1 + (7.71 + 7.49i)T + (1.92 + 66.9i)T^{2} \) |
| 71 | \( 1 + (2.28 - 0.453i)T + (65.6 - 27.0i)T^{2} \) |
| 73 | \( 1 + (10.8 - 9.38i)T + (10.4 - 72.2i)T^{2} \) |
| 79 | \( 1 + (3.89 + 2.78i)T + (25.4 + 74.7i)T^{2} \) |
| 83 | \( 1 + (2.17 + 1.37i)T + (35.5 + 74.9i)T^{2} \) |
| 89 | \( 1 + (-0.539 - 6.23i)T + (-87.6 + 15.2i)T^{2} \) |
| 97 | \( 1 + (-1.60 - 12.0i)T + (-93.6 + 25.3i)T^{2} \) |
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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
−10.82968735069999677846112043406, −10.43812227299580999839323723855, −9.629921094883457485796264965945, −8.551177833432804794665162056231, −7.69093349352432540035490634290, −7.05618691673542678543246551040, −5.80258315024962063143943954406, −4.32214627006181901873000891156, −3.72887265558424422506519180351, −2.03288152291991171641674200474,
0.43742884639022225773708923594, 1.41800753485581694645433677515, 3.91875085075286488998431539124, 4.50759442701307963013877301463, 5.54280075873051670401311773676, 7.33713471590655759080810940470, 7.59287249251420104482471644918, 8.935662044595100387406515856804, 9.033339770856142221673245336269, 10.15779302167510842695266131501
