L(s) = 1 | + (0.965 + 0.258i)2-s + (0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (2.10 + 0.754i)5-s + (0.499 − 0.866i)6-s + (−0.506 − 0.506i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.83 + 1.27i)10-s − 1.16·11-s + (0.707 − 0.707i)12-s + (6.66 − 1.78i)13-s + (−0.358 − 0.620i)14-s + (1.27 − 1.83i)15-s + (0.500 + 0.866i)16-s + (−0.465 + 1.73i)17-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.149 − 0.557i)3-s + (0.433 + 0.249i)4-s + (0.941 + 0.337i)5-s + (0.204 − 0.353i)6-s + (−0.191 − 0.191i)7-s + (0.249 + 0.249i)8-s + (−0.288 − 0.166i)9-s + (0.581 + 0.402i)10-s − 0.350·11-s + (0.204 − 0.204i)12-s + (1.84 − 0.495i)13-s + (−0.0957 − 0.165i)14-s + (0.328 − 0.474i)15-s + (0.125 + 0.216i)16-s + (−0.112 + 0.421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.62666 - 0.132385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.62666 - 0.132385i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-2.10 - 0.754i)T \) |
| 19 | \( 1 + (4.14 - 1.33i)T \) |
good | 7 | \( 1 + (0.506 + 0.506i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + (-6.66 + 1.78i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (0.465 - 1.73i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (0.517 + 1.93i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.32 + 4.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 0.506iT - 31T^{2} \) |
| 37 | \( 1 + (3.08 - 3.08i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.46 - 1.42i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.03 + 1.08i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (1.65 - 0.443i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (9.62 - 2.57i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-6.04 - 10.4i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.75 + 13.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.91 - 7.13i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (6.60 - 3.81i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.63 + 0.707i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (7.23 + 12.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.66 - 9.66i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.91 - 5.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 - 0.861i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.71539951644807292935993876277, −10.10864794242506559952280743492, −8.689506392950283065457794766919, −8.096063430982226633603841289348, −6.70750109133933492503729822456, −6.27202709025432201312125015550, −5.41201520807975726254183725771, −3.94641398226489097542368265178, −2.86231504609087759523824516157, −1.60367741052036589316876212589,
1.68464561363577982397826916018, 2.99684381844538693093074742152, 4.12446627020314056181352568837, 5.14800576641808054256439657633, 6.00553770704914323763633802795, 6.77583054797315805751474063030, 8.413041611170052053453592020607, 9.027771279524873227539135222169, 9.989844066920121709277519245783, 10.79793693940023424412276968588