L(s) = 1 | + (−1.22 − 0.710i)2-s + (1.09 − 1.09i)3-s + (0.991 + 1.73i)4-s + (−2.11 + 0.560i)6-s + 0.973i·7-s + (0.0202 − 2.82i)8-s + 0.616i·9-s + (1.40 + 1.40i)11-s + (2.97 + 0.813i)12-s + (4.60 − 4.60i)13-s + (0.691 − 1.19i)14-s + (−2.03 + 3.44i)16-s − 0.490·17-s + (0.438 − 0.754i)18-s + (4.54 − 4.54i)19-s + ⋯ |
L(s) = 1 | + (−0.864 − 0.502i)2-s + (0.630 − 0.630i)3-s + (0.495 + 0.868i)4-s + (−0.861 + 0.228i)6-s + 0.368i·7-s + (0.00714 − 0.999i)8-s + 0.205i·9-s + (0.424 + 0.424i)11-s + (0.859 + 0.234i)12-s + (1.27 − 1.27i)13-s + (0.184 − 0.318i)14-s + (−0.508 + 0.861i)16-s − 0.118·17-s + (0.103 − 0.177i)18-s + (1.04 − 1.04i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08997 - 0.543985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08997 - 0.543985i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.710i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.09 + 1.09i)T - 3iT^{2} \) |
| 7 | \( 1 - 0.973iT - 7T^{2} \) |
| 11 | \( 1 + (-1.40 - 1.40i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.60 + 4.60i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.490T + 17T^{2} \) |
| 19 | \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \) |
| 23 | \( 1 - 1.94iT - 23T^{2} \) |
| 29 | \( 1 + (3.74 - 3.74i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + (4.55 + 4.55i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (-1.79 - 1.79i)T + 43iT^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + (5.61 + 5.61i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.44 - 8.44i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.01 + 3.01i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.07 - 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.897iT - 71T^{2} \) |
| 73 | \( 1 - 9.71iT - 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + (0.815 - 0.815i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.12iT - 89T^{2} \) |
| 97 | \( 1 - 7.54T + 97T^{2} \) |
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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
−11.07377978005446662979867645519, −10.22868880820803159403264703619, −9.090127556883707601000282370734, −8.524536539665312303257429822034, −7.60879075738881650166411082057, −6.87134552323787431983406374147, −5.42930194857329474255322584535, −3.61031154101618971058326347838, −2.59416463587984407894107794533, −1.27860447325928339487701772158,
1.42203458514576947960960602343, 3.31520959799533586941644600809, 4.41838691567650338253669331563, 6.01085989980836550008462959181, 6.71029107409713533279594682856, 8.001745469447698717315220125669, 8.717696103169368520598957952422, 9.488448873263459027945930130606, 10.13769652516747445713390478369, 11.24641254275352620447822294882