L(s) = 1 | + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + (−2.59 − 6.50i)7-s − 2.82·8-s + (2.73 + 1.58i)10-s + (5.13 − 8.89i)11-s + 7.02i·13-s + (6.12 − 7.77i)14-s + (−2.00 − 3.46i)16-s + (−27.4 − 15.8i)17-s + (−26.9 + 15.5i)19-s + 4.47i·20-s + 14.5·22-s + (−11.8 − 20.5i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.370 − 0.928i)7-s − 0.353·8-s + (0.273 + 0.158i)10-s + (0.466 − 0.808i)11-s + 0.540i·13-s + (0.437 − 0.555i)14-s + (−0.125 − 0.216i)16-s + (−1.61 − 0.933i)17-s + (−1.41 + 0.818i)19-s + 0.223i·20-s + 0.660·22-s + (−0.514 − 0.891i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0172 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0172 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.102528636\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.102528636\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (2.59 + 6.50i)T \) |
good | 11 | \( 1 + (-5.13 + 8.89i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 7.02iT - 169T^{2} \) |
| 17 | \( 1 + (27.4 + 15.8i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (26.9 - 15.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.8 + 20.5i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 9.19T + 841T^{2} \) |
| 31 | \( 1 + (-17.4 - 10.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (24.0 + 41.7i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 65.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.03T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-53.6 + 30.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-0.690 + 1.19i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-95.1 - 54.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.3 - 19.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7.95 - 13.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 53.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-62.6 - 36.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (53.2 + 92.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 49.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-142. + 82.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 49.4iT - 9.40e3T^{2} \) |
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
−10.22889508563712574770900382881, −8.972130868929332815277177738943, −8.603211969941228743265100103619, −7.21578226101019333563486755598, −6.60210259018363290808216582688, −5.78841469308543314456218852730, −4.44866436769456770424905305386, −3.85412573275727623876582032825, −2.24447470898618180944206367821, −0.33419174427134459413070788457,
1.84437501232897187007020613582, 2.65231109517236286067485317558, 4.00642537591694223507966732626, 4.96793526556183784524584534934, 6.17167944664507287348468719478, 6.66639824668245659182952465207, 8.230431758457581578470142882037, 9.076616897425843381744795183573, 9.790162902948498563328713187881, 10.67874165575758158025885548593