L(s) = 1 | + (0.707 + 0.707i)3-s + (−2.23 − 0.0836i)5-s + (0.0627 + 2.64i)7-s + 1.00i·9-s − 3.98·11-s + (−0.500 − 0.500i)13-s + (−1.52 − 1.63i)15-s + (1.67 − 1.67i)17-s − 7.21·19-s + (−1.82 + 1.91i)21-s + (5.16 − 5.16i)23-s + (4.98 + 0.373i)25-s + (−0.707 + 0.707i)27-s − 3.65i·29-s − 4.93i·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.999 − 0.0373i)5-s + (0.0237 + 0.999i)7-s + 0.333i·9-s − 1.20·11-s + (−0.138 − 0.138i)13-s + (−0.392 − 0.423i)15-s + (0.407 − 0.407i)17-s − 1.65·19-s + (−0.398 + 0.417i)21-s + (1.07 − 1.07i)23-s + (0.997 + 0.0747i)25-s + (−0.136 + 0.136i)27-s − 0.678i·29-s − 0.886i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4542524740\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4542524740\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.23 + 0.0836i)T \) |
| 7 | \( 1 + (-0.0627 - 2.64i)T \) |
good | 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + (0.500 + 0.500i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.67 + 1.67i)T - 17iT^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + (-5.16 + 5.16i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.65iT - 29T^{2} \) |
| 31 | \( 1 + 4.93iT - 31T^{2} \) |
| 37 | \( 1 + (-0.292 - 0.292i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (3.65 - 3.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.305 - 0.305i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.39 + 5.39i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 - 7.11iT - 61T^{2} \) |
| 67 | \( 1 + (0.944 + 0.944i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 + (1.38 + 1.38i)T + 73iT^{2} \) |
| 79 | \( 1 + 8.64iT - 79T^{2} \) |
| 83 | \( 1 + (11.9 + 11.9i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 + (7.43 - 7.43i)T - 97iT^{2} \) |
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
−8.846260630197062602823625486653, −8.444678678547014886683421988149, −7.75894742773483494250062629977, −6.84382069170238872643508757898, −5.72085369583022399136679814578, −4.88352958711504245274476854582, −4.14356760397704045977489908542, −2.95266493243527068522046043154, −2.33823257479644297215881010444, −0.16759613966904498662745504018,
1.30892733756317159385650168993, 2.76973325902893529008630620321, 3.62903470780896998171910279158, 4.47547989334331794444073723995, 5.40767203971729299918984049729, 6.81335674615289254722060333204, 7.15041933669602068515894740316, 8.124548667924828113608949816796, 8.397014410712015120627376497063, 9.556296832680223948328622946161