L(s) = 1 | + i·2-s + (1.52 + 0.816i)3-s − 4-s + (−1.82 + 3.15i)5-s + (−0.816 + 1.52i)6-s + (−1.58 − 2.12i)7-s − i·8-s + (1.66 + 2.49i)9-s + (−3.15 − 1.82i)10-s + (4.38 − 2.53i)11-s + (−1.52 − 0.816i)12-s + (2.94 − 1.69i)13-s + (2.12 − 1.58i)14-s + (−5.35 + 3.33i)15-s + 16-s + (−0.774 + 1.34i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.881 + 0.471i)3-s − 0.5·4-s + (−0.814 + 1.41i)5-s + (−0.333 + 0.623i)6-s + (−0.598 − 0.801i)7-s − 0.353i·8-s + (0.555 + 0.831i)9-s + (−0.997 − 0.576i)10-s + (1.32 − 0.763i)11-s + (−0.440 − 0.235i)12-s + (0.816 − 0.471i)13-s + (0.566 − 0.422i)14-s + (−1.38 + 0.860i)15-s + 0.250·16-s + (−0.187 + 0.325i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.778803 + 0.892190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.778803 + 0.892190i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.52 - 0.816i)T \) |
| 7 | \( 1 + (1.58 + 2.12i)T \) |
good | 5 | \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.38 + 2.53i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.94 + 1.69i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.774 - 1.34i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.707 - 0.408i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.47 - 0.850i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.60 + 2.08i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.16iT - 31T^{2} \) |
| 37 | \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.01 - 1.76i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.06 + 5.30i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6.74T + 47T^{2} \) |
| 53 | \( 1 + (11.4 + 6.63i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.17T + 59T^{2} \) |
| 61 | \( 1 - 7.25iT - 61T^{2} \) |
| 67 | \( 1 - 2.45T + 67T^{2} \) |
| 71 | \( 1 - 6.74iT - 71T^{2} \) |
| 73 | \( 1 + (-3.76 - 2.17i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-0.768 + 1.33i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.01 + 10.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.59 + 3.23i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.05111882161516943989519073300, −13.07613608866841005648949366450, −11.28025497954490714766993275954, −10.51586918344550917803218588527, −9.328468602792257801908490429056, −8.167162597304026923735905049187, −7.18980838850759167388703518729, −6.26640073765063446594360134079, −3.92526522846699080652196257011, −3.42678836646236090500192303548,
1.54214748193136969685118606353, 3.50400897618964010540468902494, 4.62843204363667675784611840737, 6.58696251405970989556172423234, 8.136291357364964223287514342215, 9.102375474969984466050912792543, 9.378667621038746717245474313105, 11.46137484846577398864684534776, 12.34797897122208150165855478618, 12.73777738853046009045712714294