L(s) = 1 | + (3 + 1.73i)5-s + (3.5 − 6.06i)7-s + (−5 − 8.66i)11-s − 12.1i·13-s + (6 − 3.46i)17-s + (−28.5 − 16.4i)19-s + (−20 + 34.6i)23-s + (−6.5 − 11.2i)25-s − 16·29-s + (−4.5 + 2.59i)31-s + (21 − 12.1i)35-s + (−2.5 + 4.33i)37-s + 24.2i·41-s + 19·43-s + (−45 − 25.9i)47-s + ⋯ |
L(s) = 1 | + (0.600 + 0.346i)5-s + (0.5 − 0.866i)7-s + (−0.454 − 0.787i)11-s − 0.932i·13-s + (0.352 − 0.203i)17-s + (−1.5 − 0.866i)19-s + (−0.869 + 1.50i)23-s + (−0.260 − 0.450i)25-s − 0.551·29-s + (−0.145 + 0.0838i)31-s + (0.599 − 0.346i)35-s + (−0.0675 + 0.117i)37-s + 0.591i·41-s + 0.441·43-s + (−0.957 − 0.552i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.277371018\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277371018\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-3.5 + 6.06i)T \) |
good | 5 | \( 1 + (-3 - 1.73i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (5 + 8.66i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 12.1iT - 169T^{2} \) |
| 17 | \( 1 + (-6 + 3.46i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (28.5 + 16.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (20 - 34.6i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 16T + 841T^{2} \) |
| 31 | \( 1 + (4.5 - 2.59i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 24.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19T + 1.84e3T^{2} \) |
| 47 | \( 1 + (45 + 25.9i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (16 + 27.7i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-36 + 20.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-18 - 10.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-29.5 - 51.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 26T + 5.04e3T^{2} \) |
| 73 | \( 1 + (16.5 - 9.52i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-23.5 + 40.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 24.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (102 + 58.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 48.4iT - 9.40e3T^{2} \) |
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
−9.706092426804465938922936927179, −8.473091084642873491439282105183, −7.88395613136364560001714719200, −6.96157271867547547707711188689, −5.98735882340740637040964673027, −5.24066278077937149445927053177, −4.08120772698502881533513167385, −3.04399171568788945980949152030, −1.83704740083236372395238318583, −0.36601594743148967117499750050,
1.78159499552840021441762409626, 2.30301031099075463885738364115, 4.02123937540165814103663396633, 4.85432574358408452189282929330, 5.81155833086029251354234016187, 6.49098263040847651116508761766, 7.70851120182622065259180254765, 8.485371422298722147692937721614, 9.197757751204910747950942447450, 10.01081415830155946692731268955