L(s) = 1 | + (−2.02 + 1.47i)2-s + (−0.309 + 0.951i)3-s + (1.32 − 4.06i)4-s + (2.20 − 0.390i)5-s + (−0.774 − 2.38i)6-s − 7-s + (1.76 + 5.43i)8-s + (−0.809 − 0.587i)9-s + (−3.88 + 4.03i)10-s + (1.92 − 1.39i)11-s + (3.46 + 2.51i)12-s + (−2.89 − 2.10i)13-s + (2.02 − 1.47i)14-s + (−0.309 + 2.21i)15-s + (−4.65 − 3.38i)16-s + (0.830 + 2.55i)17-s + ⋯ |
L(s) = 1 | + (−1.43 + 1.04i)2-s + (−0.178 + 0.549i)3-s + (0.661 − 2.03i)4-s + (0.984 − 0.174i)5-s + (−0.316 − 0.972i)6-s − 0.377·7-s + (0.623 + 1.92i)8-s + (−0.269 − 0.195i)9-s + (−1.22 + 1.27i)10-s + (0.579 − 0.421i)11-s + (0.999 + 0.726i)12-s + (−0.802 − 0.583i)13-s + (0.541 − 0.393i)14-s + (−0.0798 + 0.571i)15-s + (−1.16 − 0.845i)16-s + (0.201 + 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.687467 + 0.166969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.687467 + 0.166969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (-2.20 + 0.390i)T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + (2.02 - 1.47i)T + (0.618 - 1.90i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 1.39i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (2.89 + 2.10i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.830 - 2.55i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.88 + 5.81i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-6.64 + 4.82i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.84 + 5.67i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.810 + 2.49i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 1.20i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.42 - 1.76i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 + (-1.12 + 3.46i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.83 - 8.73i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-12.2 - 8.93i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (4.03 - 2.93i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.07 - 6.37i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.49 - 7.66i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.06 + 4.40i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.93 + 15.1i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (5.23 + 16.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (9.43 - 6.85i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.534 + 1.64i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.40201175209037036899090968302, −9.894241748170451132373369198802, −9.026005146120718094014657555206, −8.606269572452573836373272138533, −7.29186107500245954563334076738, −6.41867639431156886798088874522, −5.76813630865291101726983491380, −4.67802660295770425460352896447, −2.61962166237157886135062854389, −0.76401571979112216140134739580,
1.30989421363427196772169530417, 2.23817930406025348482196144213, 3.38751098063189594335005008843, 5.24158908291376617704807221680, 6.68325332606862787180549634714, 7.23684889675053660892827280839, 8.402233184352400526560395379940, 9.475990584758375658600293633473, 9.644889027129911153870462735702, 10.69712945780347110798932888616