L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.22 + 0.707i)3-s + (0.499 + 0.866i)4-s + (0.965 − 0.258i)5-s + 1.41·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.965 − 0.258i)10-s + (−1.22 − 0.707i)12-s + 1.41i·13-s + (−0.999 + i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)18-s + (−0.707 + 1.22i)19-s + (0.707 + 0.707i)20-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.22 + 0.707i)3-s + (0.499 + 0.866i)4-s + (0.965 − 0.258i)5-s + 1.41·6-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−0.965 − 0.258i)10-s + (−1.22 − 0.707i)12-s + 1.41i·13-s + (−0.999 + i)15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)18-s + (−0.707 + 1.22i)19-s + (0.707 + 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5087422049\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5087422049\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - 1.41iT - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.680126677903731252675855733059, −9.053752199210276398007579869833, −8.272288429487866248881142552370, −7.05644519003377489838859051245, −6.31487462333139799479427403915, −5.69562705510544101724880455644, −4.59150190867362213011870507895, −3.90084931190201240306935607614, −2.38323199520802294766618805055, −1.38125075447638445603473234408,
0.60431606213659578211737089851, 1.80654929901132813356157066302, 2.89325841134733836415451060181, 4.92645398320260254496069707904, 5.48644615679214456594823520688, 6.21974750237787680302702286012, 6.71518558117915285973621809292, 7.46099534964262413592348233977, 8.345112607149099350141992112440, 9.229984787265873937954991398682