L(s) = 1 | + i·2-s + (−0.597 + 1.62i)3-s − 4-s − 5-s + (−1.62 − 0.597i)6-s + 3.80i·7-s − i·8-s + (−2.28 − 1.94i)9-s − i·10-s − 3.48·11-s + (0.597 − 1.62i)12-s + 3.19·13-s − 3.80·14-s + (0.597 − 1.62i)15-s + 16-s − 4.76·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.344 + 0.938i)3-s − 0.5·4-s − 0.447·5-s + (−0.663 − 0.243i)6-s + 1.43i·7-s − 0.353i·8-s + (−0.762 − 0.647i)9-s − 0.316i·10-s − 1.05·11-s + (0.172 − 0.469i)12-s + 0.887·13-s − 1.01·14-s + (0.154 − 0.419i)15-s + 0.250·16-s − 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.233878 - 0.282401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.233878 - 0.282401i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.597 - 1.62i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (-0.785 - 4.73i)T \) |
good | 7 | \( 1 - 3.80iT - 7T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 + 1.88iT - 19T^{2} \) |
| 29 | \( 1 + 7.50iT - 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 + 4.88iT - 37T^{2} \) |
| 41 | \( 1 - 3.99iT - 41T^{2} \) |
| 43 | \( 1 - 6.96iT - 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 - 7.35iT - 59T^{2} \) |
| 61 | \( 1 + 3.31iT - 61T^{2} \) |
| 67 | \( 1 - 3.70iT - 67T^{2} \) |
| 71 | \( 1 - 9.92iT - 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 10.1iT - 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 2.98iT - 97T^{2} \) |
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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
−11.22402879375091848076539790079, −10.10390197603396066936565955878, −9.129322544541121880549129847020, −8.662681701699241122438024510257, −7.74819220646292668989195653468, −6.40818318471034994327256802962, −5.65679775348252419553306219394, −4.94407703977717673641208496306, −3.86958191320295732089533238149, −2.62151847142217238909736487336,
0.20313434093206486597765129196, 1.53067880970679835454150746718, 2.97718164800625148147050918512, 4.13899347775565793975298448135, 5.13707017752271864909179542752, 6.44949177220387028487035348140, 7.25687915927897975934632828369, 8.078812222710466209393649632758, 8.835288002994581515578228382992, 10.30288096591849863633499865493