L(s) = 1 | + (0.679 − 2.53i)2-s + (−0.965 + 0.258i)3-s + (−4.23 − 2.44i)4-s + 2.62i·6-s + (−2.44 + 1.00i)7-s + (−5.35 + 5.35i)8-s + (0.866 − 0.499i)9-s + (−1.94 + 3.36i)11-s + (4.71 + 1.26i)12-s + (0.707 + 0.707i)13-s + (0.891 + 6.88i)14-s + (5.04 + 8.73i)16-s + (−1.34 − 5.02i)17-s + (−0.679 − 2.53i)18-s + (3.33 + 5.78i)19-s + ⋯ |
L(s) = 1 | + (0.480 − 1.79i)2-s + (−0.557 + 0.149i)3-s + (−2.11 − 1.22i)4-s + 1.07i·6-s + (−0.924 + 0.380i)7-s + (−1.89 + 1.89i)8-s + (0.288 − 0.166i)9-s + (−0.585 + 1.01i)11-s + (1.36 + 0.364i)12-s + (0.196 + 0.196i)13-s + (0.238 + 1.83i)14-s + (1.26 + 2.18i)16-s + (−0.326 − 1.21i)17-s + (−0.160 − 0.597i)18-s + (0.765 + 1.32i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.182712 + 0.0675728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.182712 + 0.0675728i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.44 - 1.00i)T \) |
good | 2 | \( 1 + (-0.679 + 2.53i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.94 - 3.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.34 + 5.02i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.33 - 5.78i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.20 + 1.66i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 7.76iT - 29T^{2} \) |
| 31 | \( 1 + (5.56 + 3.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.0133 + 0.0497i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 5.55iT - 41T^{2} \) |
| 43 | \( 1 + (4.97 - 4.97i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.04 + 0.547i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.540 - 2.01i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.61 + 6.26i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.47 - 0.396i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + (9.08 - 2.43i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (9.22 - 5.32i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.49 - 1.49i)T + 83iT^{2} \) |
| 89 | \( 1 + (7.46 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.34 - 3.34i)T - 97iT^{2} \) |
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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
−11.06995360083369925314836165261, −10.03009082530453043354328280697, −9.836030926975996166563075863624, −8.846123111321136093495186289473, −7.27486577741175171634621315118, −5.87715482163002916061695516790, −5.06723919883995111947472249147, −4.02608252153939603365146937344, −3.00765884349617643453648451978, −1.80265329249215595436512037591,
0.10248654286790987563387213875, 3.34503146157089871808641222018, 4.36791739805926741172475619621, 5.59808807670327860879320222457, 6.08720465164456377466598456022, 6.92258741269170114389608684642, 7.79978406519198698864538791357, 8.605159605004823011808757853688, 9.641156208182509750581255657934, 10.67089730331250581331607424969