L(s) = 1 | + (0.5 − 1.32i)2-s − 2.44·3-s + (−1.50 − 1.32i)4-s − 2·5-s + (−1.22 + 3.24i)6-s + (−2.50 + 1.32i)8-s + 2.99·9-s + (−1 + 2.64i)10-s − 2.44·11-s + (3.67 + 3.24i)12-s − 6.48i·13-s + 4.89·15-s + (0.500 + 3.96i)16-s + (1.49 − 3.96i)18-s − 5.29i·19-s + (3.00 + 2.64i)20-s + ⋯ |
L(s) = 1 | + (0.353 − 0.935i)2-s − 1.41·3-s + (−0.750 − 0.661i)4-s − 0.894·5-s + (−0.499 + 1.32i)6-s + (−0.883 + 0.467i)8-s + 0.999·9-s + (−0.316 + 0.836i)10-s − 0.738·11-s + (1.06 + 0.935i)12-s − 1.79i·13-s + 1.26·15-s + (0.125 + 0.992i)16-s + (0.353 − 0.935i)18-s − 1.21i·19-s + (0.670 + 0.591i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0403112 + 0.352771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0403112 + 0.352771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 1.32i)T \) |
| 31 | \( 1 + (4.89 - 2.64i)T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 6.48iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 - 6.48iT - 29T^{2} \) |
| 37 | \( 1 + 6.48iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 6.48iT - 53T^{2} \) |
| 59 | \( 1 - 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 6.48iT - 61T^{2} \) |
| 67 | \( 1 + 5.29iT - 67T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 - 9.79T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 12.9iT - 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−12.60431838951986644091627347345, −11.80667906172002906943949448474, −10.80135333925036004413562055908, −10.49344607819335601950750199790, −8.757639882830361925831258132082, −7.25575737661936670640808061040, −5.57956087110193676798227316260, −4.91227231007522048575047700061, −3.22837440215506083224048691802, −0.39752560345077829971982140564,
3.99672674075318482324506887584, 5.05271682383941557937653804844, 6.21650281125758753172170990757, 7.17527346878953826515586623562, 8.303937283223902428824512448569, 9.783724541813131728993301852765, 11.30515539781539659865908118981, 11.86981343051230476217696885176, 12.83113369894484413239137123985, 14.02363090910725983114108323382