| L(s) = 1 | + (0.366 − 1.36i)2-s + (−2.84 − 0.962i)3-s + (−1.73 − i)4-s + (−4.69 + 1.71i)5-s + (−2.35 + 3.52i)6-s + (−1.00 − 6.92i)7-s + (−2 + 1.99i)8-s + (7.14 + 5.46i)9-s + (0.626 + 7.04i)10-s + (8.28 + 4.78i)11-s + (3.95 + 4.50i)12-s + (−9.08 + 9.08i)13-s + (−9.83 − 1.16i)14-s + (14.9 − 0.359i)15-s + (1.99 + 3.46i)16-s + (6.55 + 24.4i)17-s + ⋯ |
| L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.947 − 0.320i)3-s + (−0.433 − 0.250i)4-s + (−0.939 + 0.343i)5-s + (−0.392 + 0.588i)6-s + (−0.143 − 0.989i)7-s + (−0.250 + 0.249i)8-s + (0.794 + 0.607i)9-s + (0.0626 + 0.704i)10-s + (0.753 + 0.434i)11-s + (0.329 + 0.375i)12-s + (−0.698 + 0.698i)13-s + (−0.702 − 0.0832i)14-s + (0.999 − 0.0239i)15-s + (0.124 + 0.216i)16-s + (0.385 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.402685 + 0.233246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.402685 + 0.233246i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.366 + 1.36i)T \) |
| 3 | \( 1 + (2.84 + 0.962i)T \) |
| 5 | \( 1 + (4.69 - 1.71i)T \) |
| 7 | \( 1 + (1.00 + 6.92i)T \) |
| good | 11 | \( 1 + (-8.28 - 4.78i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.08 - 9.08i)T - 169iT^{2} \) |
| 17 | \( 1 + (-6.55 - 24.4i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-10.4 - 18.1i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (33.1 + 8.88i)T + (458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 6.08T + 841T^{2} \) |
| 31 | \( 1 + (-7.22 - 4.17i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (14.2 + 3.80i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 55.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-57.2 - 57.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (74.4 + 19.9i)T + (1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (0.192 + 0.717i)T + (-2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-88.7 - 51.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (92.7 - 53.5i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-6.19 - 23.1i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 26.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (2.02 + 7.55i)T + (-4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (34.5 - 19.9i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (52.9 + 52.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-24.2 + 14.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (19.4 + 19.4i)T + 9.40e3iT^{2} \) |
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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
−12.07179042633468109937345495892, −11.57501789451834970401187403567, −10.43701514181553171252267382976, −9.934742260529547358629557747081, −8.114604463715267000443443150991, −7.15043483993740677827023834288, −6.14178932711898364538339157235, −4.47903862484469651988244922194, −3.78779384160033925877558731092, −1.52277830764417503504501348312,
0.29309916439646377069518473889, 3.39940986308320079192534687326, 4.82018205789561335711895756471, 5.53162302486504494198118437356, 6.76140837703708045129273627636, 7.78146330102849296360391880754, 9.013423600454613937920192774164, 9.836312779564629437560846951268, 11.45226249295778595292527171361, 11.90615041144553730612249648736
