L(s) = 1 | + (−0.618 − 1.61i)3-s + 5-s + (−0.381 − 2.61i)7-s + (−2.23 + 2.00i)9-s + 4.47i·11-s + 3.23i·13-s + (−0.618 − 1.61i)15-s + 0.763·17-s + 0.472i·19-s + (−4 + 2.23i)21-s + 4i·23-s + 25-s + (4.61 + 2.38i)27-s + 5.70i·29-s + 7.23i·31-s + ⋯ |
L(s) = 1 | + (−0.356 − 0.934i)3-s + 0.447·5-s + (−0.144 − 0.989i)7-s + (−0.745 + 0.666i)9-s + 1.34i·11-s + 0.897i·13-s + (−0.159 − 0.417i)15-s + 0.185·17-s + 0.108i·19-s + (−0.872 + 0.487i)21-s + 0.834i·23-s + 0.200·25-s + (0.888 + 0.458i)27-s + 1.05i·29-s + 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.275091706\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.275091706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 + 1.61i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (0.381 + 2.61i)T \) |
good | 11 | \( 1 - 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 - 0.472iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 5.70iT - 29T^{2} \) |
| 31 | \( 1 - 7.23iT - 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 12.9T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 2.76iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 2.76iT - 71T^{2} \) |
| 73 | \( 1 + 6.76iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 5.23iT - 97T^{2} \) |
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
−9.558929239222275767623793037209, −8.519552171502211007087415918142, −7.57716296738624212423340308636, −6.93534934127335644621101571115, −6.53863146887763680972279542725, −5.32133150645479107727527421396, −4.60861308150574951628305546906, −3.40450299375901995683680164459, −2.04396550345161337876269465063, −1.27405640238640040672855378292,
0.54148375537052537338035615179, 2.51259026344282328234957671307, 3.23089082113894238511429658890, 4.33347843379968837636844226024, 5.42289194262620725115761263088, 5.84840825225454576756091842410, 6.48829552887408984569885148206, 8.093894104887371443826677221301, 8.505229564353356643205805620792, 9.479518209220989134370121726628