| L(s) = 1 | + (0.976 − 0.413i)2-s + (1.67 − 0.428i)3-s + (−0.610 + 0.629i)4-s + (−0.817 + 0.578i)5-s + (1.46 − 1.11i)6-s + (−1.34 + 0.521i)7-s + (−1.10 + 2.84i)8-s + (2.63 − 1.43i)9-s + (−0.559 + 0.902i)10-s + (0.132 − 0.175i)11-s + (−0.754 + 1.31i)12-s + (−0.873 + 5.62i)13-s + (−1.09 + 1.06i)14-s + (−1.12 + 1.32i)15-s + (0.0455 + 1.47i)16-s + (2.01 + 0.709i)17-s + ⋯ |
| L(s) = 1 | + (0.690 − 0.292i)2-s + (0.968 − 0.247i)3-s + (−0.305 + 0.314i)4-s + (−0.365 + 0.258i)5-s + (0.596 − 0.453i)6-s + (−0.509 + 0.197i)7-s + (−0.389 + 1.00i)8-s + (0.877 − 0.479i)9-s + (−0.176 + 0.285i)10-s + (0.0400 − 0.0529i)11-s + (−0.217 + 0.380i)12-s + (−0.242 + 1.56i)13-s + (−0.293 + 0.284i)14-s + (−0.290 + 0.341i)15-s + (0.0113 + 0.369i)16-s + (0.488 + 0.172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.22769 + 0.988442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.22769 + 0.988442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.67 + 0.428i)T \) |
| 103 | \( 1 + (-9.90 + 2.22i)T \) |
| good | 2 | \( 1 + (-0.976 + 0.413i)T + (1.39 - 1.43i)T^{2} \) |
| 5 | \( 1 + (0.817 - 0.578i)T + (1.66 - 4.71i)T^{2} \) |
| 7 | \( 1 + (1.34 - 0.521i)T + (5.17 - 4.71i)T^{2} \) |
| 11 | \( 1 + (-0.132 + 0.175i)T + (-3.01 - 10.5i)T^{2} \) |
| 13 | \( 1 + (0.873 - 5.62i)T + (-12.3 - 3.94i)T^{2} \) |
| 17 | \( 1 + (-2.01 - 0.709i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (-8.36 + 0.515i)T + (18.8 - 2.33i)T^{2} \) |
| 23 | \( 1 + (-0.580 - 4.68i)T + (-22.3 + 5.61i)T^{2} \) |
| 29 | \( 1 + (0.339 - 0.737i)T + (-18.8 - 22.0i)T^{2} \) |
| 31 | \( 1 + (-3.08 + 0.0949i)T + (30.9 - 1.90i)T^{2} \) |
| 37 | \( 1 + (3.34 + 3.66i)T + (-3.41 + 36.8i)T^{2} \) |
| 41 | \( 1 + (4.50 + 0.417i)T + (40.3 + 7.53i)T^{2} \) |
| 43 | \( 1 + (1.56 + 4.92i)T + (-35.0 + 24.8i)T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + (2.93 + 4.42i)T + (-20.6 + 48.8i)T^{2} \) |
| 59 | \( 1 + (0.383 - 0.989i)T + (-43.6 - 39.7i)T^{2} \) |
| 61 | \( 1 + (-2.11 - 2.46i)T + (-9.35 + 60.2i)T^{2} \) |
| 67 | \( 1 + (-4.00 + 0.621i)T + (63.8 - 20.3i)T^{2} \) |
| 71 | \( 1 + (6.83 + 4.83i)T + (23.5 + 66.9i)T^{2} \) |
| 73 | \( 1 + (1.68 + 0.155i)T + (71.7 + 13.4i)T^{2} \) |
| 79 | \( 1 + (11.2 - 7.96i)T + (26.2 - 74.5i)T^{2} \) |
| 83 | \( 1 + (-8.92 - 1.38i)T + (79.0 + 25.1i)T^{2} \) |
| 89 | \( 1 + (-3.82 + 13.4i)T + (-75.6 - 46.8i)T^{2} \) |
| 97 | \( 1 + (-7.93 + 9.26i)T + (-14.8 - 95.8i)T^{2} \) |
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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
−9.880526854179719781616448945019, −9.314099524321124677555173602735, −8.611289261619501891355370188212, −7.53226529508774416073727483211, −7.05890321591149468231283695406, −5.69939775011269934538027929720, −4.60751410064772417336151872030, −3.53618038540868482403988598003, −3.17042259355052652299316468596, −1.80981080630990268991072445068,
0.884739658537573169224542713145, 2.95928433799511520372728651453, 3.54457441903380863185052805083, 4.67251228463237287770488431009, 5.33964800984106699838868874694, 6.47791406913469552203787638664, 7.53463528940285005130492796460, 8.188077275741469022761450759869, 9.184128483832726989907093370333, 10.07338441063071839621027768198
