L(s) = 1 | + (−0.939 − 0.342i)3-s + (0.939 − 1.62i)7-s + (0.766 + 0.642i)9-s + (−0.326 + 0.118i)13-s + (−0.5 + 0.866i)19-s + (−1.43 + 1.20i)21-s + (−0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + (−0.766 + 1.32i)31-s + 1.53·37-s + 0.347·39-s + (0.266 + 1.50i)43-s + (−1.26 − 2.19i)49-s + (0.766 − 0.642i)57-s + (0.0603 − 0.342i)61-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)3-s + (0.939 − 1.62i)7-s + (0.766 + 0.642i)9-s + (−0.326 + 0.118i)13-s + (−0.5 + 0.866i)19-s + (−1.43 + 1.20i)21-s + (−0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + (−0.766 + 1.32i)31-s + 1.53·37-s + 0.347·39-s + (0.266 + 1.50i)43-s + (−1.26 − 2.19i)49-s + (0.766 − 0.642i)57-s + (0.0603 − 0.342i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5965716834\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5965716834\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
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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
−12.27241640337287264413991556277, −11.22705388470360905952556821228, −10.68841063034004938204044037024, −9.755476444356326887659613267974, −7.990945917030257619542676595666, −7.38857740287814404294217850660, −6.28481136834883854916484166649, −4.95923144033880540540075629516, −4.02634525145321782775232691616, −1.50982751497776689218005851643,
2.27874071348170025039357565269, 4.34081692186315113372638408770, 5.39321199046958206380717898193, 6.11849197904606841331446093956, 7.58635044232470429388292242994, 8.797824659548627564189731178324, 9.644868190922176379766015402785, 10.90268483140256547162961089141, 11.62908781533890540191818559849, 12.24178980894681560527179094947