L(s) = 1 | + (0.5 − 0.866i)2-s + (0.909 − 1.47i)3-s + (−0.499 − 0.866i)4-s + (3.26 + 1.88i)5-s + (−0.821 − 1.52i)6-s + (0.385 + 2.61i)7-s − 0.999·8-s + (−1.34 − 2.68i)9-s + (3.26 − 1.88i)10-s + (−1.89 − 3.27i)11-s + (−1.73 − 0.0505i)12-s + (2.90 + 2.13i)13-s + (2.45 + 0.975i)14-s + (5.74 − 3.09i)15-s + (−0.5 + 0.866i)16-s + (1.16 + 2.00i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.525 − 0.851i)3-s + (−0.249 − 0.433i)4-s + (1.45 + 0.842i)5-s + (−0.335 − 0.622i)6-s + (0.145 + 0.989i)7-s − 0.353·8-s + (−0.448 − 0.893i)9-s + (1.03 − 0.595i)10-s + (−0.570 − 0.988i)11-s + (−0.499 − 0.0146i)12-s + (0.805 + 0.593i)13-s + (0.657 + 0.260i)14-s + (1.48 − 0.799i)15-s + (−0.125 + 0.216i)16-s + (0.281 + 0.487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.05646 - 1.35983i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05646 - 1.35983i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.909 + 1.47i)T \) |
| 7 | \( 1 + (-0.385 - 2.61i)T \) |
| 13 | \( 1 + (-2.90 - 2.13i)T \) |
good | 5 | \( 1 + (-3.26 - 1.88i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.89 + 3.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 2.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.00 + 5.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.79 + 3.34i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.07iT - 29T^{2} \) |
| 31 | \( 1 + (-4.94 - 8.56i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.92 + 3.42i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.69iT - 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 + (6.71 + 3.87i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.653 + 0.377i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.68 - 2.12i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.37 + 4.83i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.51 - 2.60i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + (-2.53 - 4.38i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.402 - 0.696i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.88iT - 83T^{2} \) |
| 89 | \( 1 + (-13.5 - 7.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.04T + 97T^{2} \) |
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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
−10.72360914301473252986936975039, −9.779328685925993994111645037918, −8.919328157196524409063043977281, −8.210721168399729724682386130603, −6.59273423208169128162719319358, −6.16181883137145983903109223851, −5.24929604183716230601915256273, −3.27347504282539491136884170159, −2.56223916858704002214978240064, −1.60860392795881047097910626036,
1.77778473422348816436162015436, 3.40649685220700708369394320192, 4.53705139314172236173716707087, 5.31199326193676026118822315591, 6.08134830497399155093803643907, 7.60011118168379650258269657132, 8.198981765723430840047854245160, 9.368103330674595813916486646461, 9.966114359102146019134328465089, 10.48003735252469025719820413850