L(s) = 1 | + 2-s + (0.403 − 1.68i)3-s + 4-s + (−1.80 + 1.04i)5-s + (0.403 − 1.68i)6-s + (−0.800 − 2.52i)7-s + 8-s + (−2.67 − 1.35i)9-s + (−1.80 + 1.04i)10-s + (−1.07 − 1.86i)11-s + (0.403 − 1.68i)12-s + (0.217 − 3.59i)13-s + (−0.800 − 2.52i)14-s + (1.02 + 3.45i)15-s + 16-s + 0.557·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.232 − 0.972i)3-s + 0.5·4-s + (−0.806 + 0.465i)5-s + (0.164 − 0.687i)6-s + (−0.302 − 0.953i)7-s + 0.353·8-s + (−0.891 − 0.452i)9-s + (−0.570 + 0.329i)10-s + (−0.325 − 0.563i)11-s + (0.116 − 0.486i)12-s + (0.0602 − 0.998i)13-s + (−0.213 − 0.673i)14-s + (0.264 + 0.892i)15-s + 0.250·16-s + 0.135·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.999641 - 1.46295i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.999641 - 1.46295i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.403 + 1.68i)T \) |
| 7 | \( 1 + (0.800 + 2.52i)T \) |
| 13 | \( 1 + (-0.217 + 3.59i)T \) |
good | 5 | \( 1 + (1.80 - 1.04i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.07 + 1.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 0.557T + 17T^{2} \) |
| 19 | \( 1 + (-1.94 + 3.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.07iT - 23T^{2} \) |
| 29 | \( 1 + (-6.10 - 3.52i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.21 + 5.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.31iT - 37T^{2} \) |
| 41 | \( 1 + (-0.532 - 0.307i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 - 7.50i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.507 + 0.292i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.68 + 3.85i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 1.56iT - 59T^{2} \) |
| 61 | \( 1 + (7.10 + 4.10i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 + 7.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.52 - 11.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.198 + 0.344i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.73 - 9.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 8.70iT - 89T^{2} \) |
| 97 | \( 1 + (-1.64 - 2.85i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.92894873840283266041071458842, −9.810110924532402722902021750277, −8.255882027470197735508313720404, −7.74378788045663015581624149269, −6.88496552347244603045829530593, −6.13830908900392469875640456507, −4.81760098554538584362504778788, −3.44199790356811513541871081964, −2.85408826733103076549138653957, −0.802745240387891400876195489918,
2.32453977147502063258826147385, 3.55220625453447954156892757416, 4.41878067426002771572710130971, 5.23799470702911051499159612884, 6.23522554685172561725064627240, 7.58809608306316572900802857716, 8.497300561358890889307964434566, 9.324406354859250467895768715811, 10.19442732994135212371785013602, 11.22536072823333383581796205077