L(s) = 1 | + 3-s + (1 + 1.73i)4-s + (2.5 + 0.866i)7-s + 9-s − 6·11-s + (1 + 1.73i)12-s + (1 − 3.46i)13-s + (−1.99 + 3.46i)16-s + (3 + 5.19i)17-s − 19-s + (2.5 + 0.866i)21-s + (3 − 5.19i)23-s + (2.5 − 4.33i)25-s + 27-s + (1.00 + 5.19i)28-s + (−3 − 5.19i)29-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (0.5 + 0.866i)4-s + (0.944 + 0.327i)7-s + 0.333·9-s − 1.80·11-s + (0.288 + 0.499i)12-s + (0.277 − 0.960i)13-s + (−0.499 + 0.866i)16-s + (0.727 + 1.26i)17-s − 0.229·19-s + (0.545 + 0.188i)21-s + (0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + 0.192·27-s + (0.188 + 0.981i)28-s + (−0.557 − 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62453 + 0.536125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62453 + 0.536125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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
−12.29091137516407948144273983484, −10.76336730097982592066630913679, −10.51810372628278857101993379866, −8.694531887478948531156843407273, −8.078030100597531039402791895284, −7.55690675277842499043507175633, −5.98026603602669110992264905323, −4.71183318939704073392095496173, −3.23221464868058999456757382845, −2.21528582487949343099019962483,
1.58228865631821683910614440088, 2.94397027629789813849841049066, 4.79153412721784858462342708317, 5.53585830801100031380296469818, 7.18651121217718707186110819940, 7.69134143290365378570083879270, 9.055100404151577712676096623760, 9.925866125853771218934926643199, 10.99182368036093911821271645413, 11.41713663638587335348598029106