L(s) = 1 | + (0.5 + 0.866i)2-s + (1.70 − 2.95i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 3.41·6-s − 0.999·8-s + (−4.32 − 7.49i)9-s + (−0.499 + 0.866i)10-s + (0.414 − 0.717i)11-s + (1.70 + 2.95i)12-s + 4.82·13-s + 3.41·15-s + (−0.5 − 0.866i)16-s + (1.29 − 2.23i)17-s + (4.32 − 7.49i)18-s + (0.292 + 0.507i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.985 − 1.70i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 1.39·6-s − 0.353·8-s + (−1.44 − 2.49i)9-s + (−0.158 + 0.273i)10-s + (0.124 − 0.216i)11-s + (0.492 + 0.853i)12-s + 1.33·13-s + 0.881·15-s + (−0.125 − 0.216i)16-s + (0.313 − 0.543i)17-s + (1.02 − 1.76i)18-s + (0.0671 + 0.116i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16654 - 0.819177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16654 - 0.819177i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.70 + 2.95i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + (-1.29 + 2.23i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.292 - 0.507i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.585 - 1.01i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + (1.41 - 2.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.82 - 6.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + (-2.58 - 4.47i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.24 - 5.61i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.29 + 7.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.65 - 8.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.828 - 1.43i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + (4.70 - 8.15i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.41 - 5.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 + (-6.36 - 11.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−11.20679124546015393047987559714, −9.556113259788959853087166878582, −8.690580618965066135939162608650, −8.002900944358438785974253398825, −7.16751078163214893905044444664, −6.43907643013795437640422460940, −5.68340774843600611742431850792, −3.69290326412119320768507752055, −2.81937062431897054843185001886, −1.35206151644522292437029881430,
2.06322974132694723791281788004, 3.43478377037783289120327412180, 4.02401657055136865497949621145, 5.04045434998193302857447674029, 5.94897513578361819196048530235, 7.894841351525999882793926801675, 8.805832544286284649645482947186, 9.320655348404245836041710453272, 10.20756632135622319350515757613, 10.83939874990452335718033234204