L(s) = 1 | − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s + i·8-s − 1.00i·9-s + (0.707 + 0.707i)10-s + (0.707 − 0.707i)12-s − 1.41·13-s − 1.00i·15-s + 16-s + i·17-s − 1.00·18-s + (0.707 − 0.707i)20-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s + i·8-s − 1.00i·9-s + (0.707 + 0.707i)10-s + (0.707 − 0.707i)12-s − 1.41·13-s − 1.00i·15-s + 16-s + i·17-s − 1.00·18-s + (0.707 − 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2638951065\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2638951065\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 - iT \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (-1 + i)T - iT^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + 2T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + 1.41T + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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
−9.398403663443037520932400658780, −8.324621270291112914176380621175, −7.64156959579373994791119257471, −6.48648700368981223430835552726, −5.70674229558274230796762933026, −4.52672796658609031094945337967, −4.20218366760448872586251997121, −3.17741951227808935812566039568, −2.20995232501758188853121378809, −0.22839067864915805882443768634,
1.24576847235319261336291968950, 2.98446240989403607976166261642, 4.51092979352477532666482874894, 4.85172087290494664931167109406, 5.68699881717952081235481234994, 6.61779974040013714077697731415, 7.33204829265019901056018844535, 7.88270782653415849758624214005, 8.473453977146309721753107637057, 9.633285118402863566022947473504