L(s) = 1 | + 1.41·2-s + 1.43·3-s + 2.00·4-s − 2.23i·5-s + 2.03·6-s − 10.1i·7-s + 2.82·8-s − 6.93·9-s − 3.16i·10-s − 13.0i·11-s + 2.87·12-s + 23.2·13-s − 14.4i·14-s − 3.21i·15-s + 4.00·16-s + 28.2i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.479·3-s + 0.500·4-s − 0.447i·5-s + 0.339·6-s − 1.45i·7-s + 0.353·8-s − 0.770·9-s − 0.316i·10-s − 1.18i·11-s + 0.239·12-s + 1.78·13-s − 1.02i·14-s − 0.214i·15-s + 0.250·16-s + 1.66i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.43502 - 1.11376i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.43502 - 1.11376i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (17.3 - 15.0i)T \) |
good | 3 | \( 1 - 1.43T + 9T^{2} \) |
| 7 | \( 1 + 10.1iT - 49T^{2} \) |
| 11 | \( 1 + 13.0iT - 121T^{2} \) |
| 13 | \( 1 - 23.2T + 169T^{2} \) |
| 17 | \( 1 - 28.2iT - 289T^{2} \) |
| 19 | \( 1 - 11.6iT - 361T^{2} \) |
| 29 | \( 1 - 42.4T + 841T^{2} \) |
| 31 | \( 1 - 18.7T + 961T^{2} \) |
| 37 | \( 1 - 1.14iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 72.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.96iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 0.813T + 2.20e3T^{2} \) |
| 53 | \( 1 + 26.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 94.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 74.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 80.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 83.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 8.98T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 94.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 2.32iT - 9.40e3T^{2} \) |
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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
−11.84749236509897044628502727162, −10.93244005268378726604430520954, −10.21125018685290030263036431357, −8.367122512689244958946541652077, −8.264630350181920620896956393389, −6.49440339297643145684994836184, −5.71336164799353883823464019089, −3.98741456362268215446203569484, −3.45069279993252010704355840805, −1.27586641431590126930787814755,
2.29264505855285128930203618846, 3.15341868216811998201284442434, 4.76733668683285657280481742888, 5.90761910066620950344980482915, 6.83570676272023224198915788654, 8.285139031527876226081664011528, 9.019707283255461302075919399430, 10.24010233197330744404772012364, 11.63247036121648038370055364422, 11.88160744099560859767059054504