L(s) = 1 | + 1.47i·2-s + 3.55i·3-s + 1.82·4-s − 5.24·6-s + 7.05·7-s + 8.58i·8-s − 3.65·9-s + 2.24·11-s + 6.50i·12-s + 6.86i·13-s + 10.4i·14-s − 5.34·16-s + 11.1·17-s − 5.38i·18-s + (−15.8 − 10.4i)19-s + ⋯ |
L(s) = 1 | + 0.736i·2-s + 1.18i·3-s + 0.457·4-s − 0.873·6-s + 1.00·7-s + 1.07i·8-s − 0.406·9-s + 0.203·11-s + 0.542i·12-s + 0.527i·13-s + 0.743i·14-s − 0.333·16-s + 0.658·17-s − 0.299i·18-s + (−0.836 − 0.547i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.354772252\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.354772252\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (15.8 + 10.4i)T \) |
good | 2 | \( 1 - 1.47iT - 4T^{2} \) |
| 3 | \( 1 - 3.55iT - 9T^{2} \) |
| 7 | \( 1 - 7.05T + 49T^{2} \) |
| 11 | \( 1 - 2.24T + 121T^{2} \) |
| 13 | \( 1 - 6.86iT - 169T^{2} \) |
| 17 | \( 1 - 11.1T + 289T^{2} \) |
| 23 | \( 1 - 21.1T + 529T^{2} \) |
| 29 | \( 1 - 45.9iT - 841T^{2} \) |
| 31 | \( 1 + 14.7iT - 961T^{2} \) |
| 37 | \( 1 + 25.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 64.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 44.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 34.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 20.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 79.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 26.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 99.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 109. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 124.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 85.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 35.5iT - 9.40e3T^{2} \) |
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
−11.01260628113130890992174345271, −10.43486304221770451293044122401, −9.201429500572353705876376619192, −8.521313273071369178528947254784, −7.44939142933781364290082294307, −6.59910857796619143252167224542, −5.28460906328802391605889252251, −4.75712496902479508528213899660, −3.46378074741973852633848489422, −1.85850727841953326474215101378,
1.04534661875779085101655170841, 1.85949898081057164523163992167, 3.02724664550269427629121599687, 4.48482078704781317849860303796, 5.94197236225635084579940394731, 6.80502708340035394162200628155, 7.73695714378427488775239330088, 8.339651617677013124892709128072, 9.828288725764655794904223519651, 10.60378011115524084409274764381