L(s) = 1 | − i·2-s + 1.73i·3-s − 4-s − 1.73·5-s + 1.73·6-s + (2 − 1.73i)7-s + i·8-s − 2.99·9-s + 1.73i·10-s − 1.73i·12-s + 1.73i·13-s + (−1.73 − 2i)14-s − 2.99i·15-s + 16-s − 3.46·17-s + 2.99i·18-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.999i·3-s − 0.5·4-s − 0.774·5-s + 0.707·6-s + (0.755 − 0.654i)7-s + 0.353i·8-s − 0.999·9-s + 0.547i·10-s − 0.499i·12-s + 0.480i·13-s + (−0.462 − 0.534i)14-s − 0.774i·15-s + 0.250·16-s − 0.840·17-s + 0.707i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.203029 + 0.444411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.203029 + 0.444411i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| 23 | \( 1 - iT \) |
good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 29 | \( 1 + 9iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 8.66T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 12.1iT - 97T^{2} \) |
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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
−10.29830337512985814919375355078, −9.824722748357661531068942393906, −8.565405649765283100892921738951, −8.238889663770030904643678950823, −7.07850844113640493192684156298, −5.72506712664060025412618134921, −4.63962469859591386777720407666, −4.09279397665848444004050171896, −3.29809003246557298704333918087, −1.73782153158932564270454481239,
0.22201275981737702551831725562, 1.96982109007826850296924808870, 3.26446235748117919696487282760, 4.70410686785791610149054076297, 5.41047220927664224654783423445, 6.53404776537550041285097144429, 7.16949163931782490797876191436, 8.045722799724118625413908132491, 8.534914062152172133142293588109, 9.290365232294858221586791552003