L(s) = 1 | + (−0.273 + 1.71i)3-s + (−1.33 + 2.31i)5-s + (−0.581 + 2.58i)7-s + (−2.84 − 0.937i)9-s + (0.682 + 1.18i)11-s + (−2.75 − 4.77i)13-s + (−3.58 − 2.91i)15-s + (−1.23 + 2.14i)17-s + (2.19 + 3.80i)19-s + (−4.25 − 1.70i)21-s + (−2.34 + 4.06i)23-s + (−1.05 − 1.83i)25-s + (2.38 − 4.61i)27-s + (2.94 − 5.10i)29-s − 3.11·31-s + ⋯ |
L(s) = 1 | + (−0.158 + 0.987i)3-s + (−0.596 + 1.03i)5-s + (−0.219 + 0.975i)7-s + (−0.949 − 0.312i)9-s + (0.205 + 0.356i)11-s + (−0.764 − 1.32i)13-s + (−0.925 − 0.752i)15-s + (−0.300 + 0.520i)17-s + (0.503 + 0.872i)19-s + (−0.928 − 0.371i)21-s + (−0.488 + 0.846i)23-s + (−0.211 − 0.366i)25-s + (0.458 − 0.888i)27-s + (0.547 − 0.948i)29-s − 0.559·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5599208103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5599208103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.273 - 1.71i)T \) |
| 7 | \( 1 + (0.581 - 2.58i)T \) |
good | 5 | \( 1 + (1.33 - 2.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.682 - 1.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.75 + 4.77i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.23 - 2.14i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.19 - 3.80i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.34 - 4.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.94 + 5.10i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 + (3.15 + 5.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.38 - 2.40i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.87 + 8.45i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + (1.47 - 2.56i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 - 1.32T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.11 - 1.93i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + (5.15 - 8.93i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.73 - 13.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.55 - 4.42i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.42546307867436570799090770110, −9.817719021112526423201916902793, −8.969214767939568012234825871658, −7.977066481552692247569622721703, −7.23188756109606066062997530299, −5.92029052885951045498076790619, −5.50145129093995790534758723465, −4.16181329594816081707491715037, −3.32186716171276708267118332640, −2.46682460369661613396091096800,
0.26753456463398444748549871944, 1.39152894402832773386831356686, 2.86985609917562528553632868259, 4.31890017380979056598334939306, 4.87977261918445430770940173149, 6.22551371448144741150661100900, 7.09896284266747233155476914555, 7.54285989705362854754612941565, 8.700912639686529143445857287074, 9.107143018665536669438830955548