| L(s) = 1 | + (0.415 − 1.55i)2-s + (0.0778 − 2.99i)3-s + (1.22 + 0.709i)4-s + (4.73 − 1.60i)5-s + (−4.62 − 1.36i)6-s + (3.98 + 5.75i)7-s + (6.15 − 6.15i)8-s + (−8.98 − 0.466i)9-s + (−0.529 − 8.01i)10-s + (−11.9 − 6.87i)11-s + (2.22 − 3.62i)12-s + (−13.4 + 13.4i)13-s + (10.5 − 3.78i)14-s + (−4.45 − 14.3i)15-s + (−4.15 − 7.20i)16-s + (3.17 + 11.8i)17-s + ⋯ |
| L(s) = 1 | + (0.207 − 0.776i)2-s + (0.0259 − 0.999i)3-s + (0.307 + 0.177i)4-s + (0.946 − 0.321i)5-s + (−0.770 − 0.228i)6-s + (0.568 + 0.822i)7-s + (0.769 − 0.769i)8-s + (−0.998 − 0.0518i)9-s + (−0.0529 − 0.801i)10-s + (−1.08 − 0.624i)11-s + (0.185 − 0.302i)12-s + (−1.03 + 1.03i)13-s + (0.756 − 0.270i)14-s + (−0.297 − 0.954i)15-s + (−0.259 − 0.450i)16-s + (0.186 + 0.696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0436 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0436 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28369 - 1.34104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28369 - 1.34104i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0778 + 2.99i)T \) |
| 5 | \( 1 + (-4.73 + 1.60i)T \) |
| 7 | \( 1 + (-3.98 - 5.75i)T \) |
| good | 2 | \( 1 + (-0.415 + 1.55i)T + (-3.46 - 2i)T^{2} \) |
| 11 | \( 1 + (11.9 + 6.87i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (13.4 - 13.4i)T - 169iT^{2} \) |
| 17 | \( 1 + (-3.17 - 11.8i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (0.412 + 0.714i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-5.43 - 1.45i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 4.58T + 841T^{2} \) |
| 31 | \( 1 + (-4.73 - 2.73i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-41.2 - 11.0i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 68.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (24.3 + 24.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-24.0 - 6.45i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-3.61 - 13.4i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-42.3 - 24.4i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (11.4 - 6.59i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (6.80 + 25.3i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-1.34 - 5.01i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-49.9 + 28.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (105. + 105. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (62.2 - 35.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-89.5 - 89.5i)T + 9.40e3iT^{2} \) |
| 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
−13.06915542255423536380639262016, −12.23898093623746186746553831676, −11.47130647247943810501649847486, −10.28686113008737288859325207376, −8.871911779888420208941591943518, −7.72383682030654693088569004362, −6.37427336365389694040287095872, −5.11286702864573006444977399786, −2.69524896576489589297148726832, −1.77196237713436348033203137850,
2.59365852009986732705263914721, 4.86847739803589948712203965861, 5.47635091349165255601170211670, 7.05491547867553618509762276274, 8.056114241559746516637609442176, 9.915746569582571944397938219398, 10.32624213495704192433415374417, 11.34275153333647002508339890614, 13.14706967548899494041374823940, 14.19928156728091268631729224104
