L(s) = 1 | + (0.409 − 0.409i)3-s + (−0.738 − 2.54i)7-s + 2.66i·9-s − 3.54·11-s + (2.95 − 2.95i)13-s + (5.59 + 5.59i)17-s + 3.59·19-s + (−1.34 − 0.737i)21-s + (0.0472 + 0.0472i)23-s + (2.31 + 2.31i)27-s − 5.34i·29-s − 10.3i·31-s + (−1.45 + 1.45i)33-s + (7.80 − 7.80i)37-s − 2.41i·39-s + ⋯ |
L(s) = 1 | + (0.236 − 0.236i)3-s + (−0.278 − 0.960i)7-s + 0.888i·9-s − 1.07·11-s + (0.818 − 0.818i)13-s + (1.35 + 1.35i)17-s + 0.824·19-s + (−0.292 − 0.160i)21-s + (0.00986 + 0.00986i)23-s + (0.446 + 0.446i)27-s − 0.993i·29-s − 1.86i·31-s + (−0.252 + 0.252i)33-s + (1.28 − 1.28i)37-s − 0.386i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.719687131\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.719687131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.738 + 2.54i)T \) |
good | 3 | \( 1 + (-0.409 + 0.409i)T - 3iT^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 + (-2.95 + 2.95i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.59 - 5.59i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 + (-0.0472 - 0.0472i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 + (-7.80 + 7.80i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.63iT - 41T^{2} \) |
| 43 | \( 1 + (6.93 + 6.93i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.44 - 3.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.646 - 0.646i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 + 3.51iT - 61T^{2} \) |
| 67 | \( 1 + (1.70 - 1.70i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.54T + 71T^{2} \) |
| 73 | \( 1 + (-2.43 + 2.43i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.72iT - 79T^{2} \) |
| 83 | \( 1 + (-2.04 + 2.04i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.18T + 89T^{2} \) |
| 97 | \( 1 + (-6.23 - 6.23i)T + 97iT^{2} \) |
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
−9.621748565388081932493790602849, −8.280948542792816209699383448600, −7.83138940779907380118474075323, −7.35194640469776363328881829341, −5.94239276663757659033358769722, −5.48196612332286541156226161851, −4.16665069375009264714968058863, −3.35810618630316919789812286091, −2.20678894055581145469703289454, −0.78881162509175861958524173471,
1.23454443520259453885429452406, 2.94881275821571085214569526450, 3.25054056460710128628334812204, 4.76729555446793973041333073566, 5.46900486439885388537038148742, 6.39447043453954626009005203536, 7.22268290096802069844554358288, 8.270745392616362528798140834628, 8.915449333609067074380850509871, 9.673557303448099812357945826040