L(s) = 1 | + (−1.39 − 0.245i)3-s + (0.766 + 0.642i)5-s + (−0.5 − 0.866i)7-s + (0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.909 − 1.08i)15-s + (−0.939 + 0.342i)17-s + (0.483 + 1.32i)21-s + (0.483 − 1.32i)29-s + (−0.909 + 1.08i)33-s + (0.173 − 0.984i)35-s − 1.41i·37-s + (1.39 + 0.245i)41-s + (−0.766 − 0.642i)43-s + (0.5 + 0.866i)45-s + ⋯ |
L(s) = 1 | + (−1.39 − 0.245i)3-s + (0.766 + 0.642i)5-s + (−0.5 − 0.866i)7-s + (0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.909 − 1.08i)15-s + (−0.939 + 0.342i)17-s + (0.483 + 1.32i)21-s + (0.483 − 1.32i)29-s + (−0.909 + 1.08i)33-s + (0.173 − 0.984i)35-s − 1.41i·37-s + (1.39 + 0.245i)41-s + (−0.766 − 0.642i)43-s + (0.5 + 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6673906321\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6673906321\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (1.39 + 0.245i)T + (0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.483 + 1.32i)T + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + (-1.39 - 0.245i)T + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (0.483 + 1.32i)T + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.909 + 1.08i)T + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
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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
−9.781137659678641086621682549395, −8.934592128744038169049288963920, −7.69741207889961849345985717651, −6.76783754591379395899542905418, −6.24808140329642364485493862594, −5.82047701829671240042173843896, −4.59574672922964405487864547221, −3.63410426875470771534757835836, −2.26110267207529236848524944744, −0.69686514898170891438626966884,
1.40723721851488388117922223271, 2.70470684714235883219210567213, 4.33462867664945612157678546189, 5.02948854538337432739157826806, 5.70686877060110611354578412320, 6.42313497283987295611701349371, 7.08591516916132318928023560250, 8.555390456851961774898802769075, 9.272810592429395523269311773580, 9.801243897192051595416805490752