L(s) = 1 | − 3.28·2-s + 1.73·3-s + 6.77·4-s − 2.78i·5-s − 5.68·6-s − 2.64i·7-s − 9.10·8-s + 2.99·9-s + 9.14i·10-s − 7.34i·11-s + 11.7·12-s + 3.74·13-s + 8.68i·14-s − 4.82i·15-s + 2.77·16-s − 19.4i·17-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 0.577·3-s + 1.69·4-s − 0.557i·5-s − 0.947·6-s − 0.377i·7-s − 1.13·8-s + 0.333·9-s + 0.914i·10-s − 0.667i·11-s + 0.977·12-s + 0.287·13-s + 0.620i·14-s − 0.321i·15-s + 0.173·16-s − 1.14i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.831i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.554 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6857170562\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6857170562\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (19.1 + 12.7i)T \) |
good | 2 | \( 1 + 3.28T + 4T^{2} \) |
| 5 | \( 1 + 2.78iT - 25T^{2} \) |
| 11 | \( 1 + 7.34iT - 121T^{2} \) |
| 13 | \( 1 - 3.74T + 169T^{2} \) |
| 17 | \( 1 + 19.4iT - 289T^{2} \) |
| 19 | \( 1 - 7.40iT - 361T^{2} \) |
| 29 | \( 1 + 40.6T + 841T^{2} \) |
| 31 | \( 1 - 8.80T + 961T^{2} \) |
| 37 | \( 1 - 22.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 0.856T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 34.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 27.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 94.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 61.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 77.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 132.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 124. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 59.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 66.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 7.24iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18408514373517814692755184749, −9.400680526147163030209203024503, −8.735955816913072202615127061340, −8.008690957966841087628994442161, −7.26616380576844075590291050344, −6.15437032126719537175157360528, −4.61584244162593869712025338004, −3.12299858154916048613105014395, −1.69104636152786412915925381595, −0.45061815261414706631999427963,
1.57595183314469183349567824770, 2.58293067252763883078338701483, 4.03407098862599312602417050085, 5.84056321157962331363406650605, 6.98862800670089780718543017765, 7.61524961228862988310998067764, 8.569190836470611968040761380047, 9.178045483657519949365877458348, 10.08347467570037695149669514761, 10.67713454765576905423457161536