L(s) = 1 | + (−1.06 − 1.06i)2-s + (1.36 − 1.06i)3-s + 0.267i·4-s + (−1.06 − 1.06i)5-s + (−2.58 − 0.320i)6-s + (−1 − i)7-s + (−1.84 + 1.84i)8-s + (0.732 − 2.90i)9-s + 2.26i·10-s + (2.90 − 2.90i)11-s + (0.285 + 0.366i)12-s + 2.12i·14-s + (−2.58 − 0.320i)15-s + 4.46·16-s − 5.03·17-s + (−3.87 + 2.31i)18-s + ⋯ |
L(s) = 1 | + (−0.752 − 0.752i)2-s + (0.788 − 0.614i)3-s + 0.133i·4-s + (−0.476 − 0.476i)5-s + (−1.05 − 0.130i)6-s + (−0.377 − 0.377i)7-s + (−0.652 + 0.652i)8-s + (0.244 − 0.969i)9-s + 0.717i·10-s + (0.877 − 0.877i)11-s + (0.0823 + 0.105i)12-s + 0.569i·14-s + (−0.668 − 0.0827i)15-s + 1.11·16-s − 1.22·17-s + (−0.913 + 0.546i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.141569 + 0.840813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.141569 + 0.840813i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.36 + 1.06i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.06 + 1.06i)T + 2iT^{2} \) |
| 5 | \( 1 + (1.06 + 1.06i)T + 5iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.90 + 2.90i)T - 11iT^{2} \) |
| 17 | \( 1 + 5.03T + 17T^{2} \) |
| 19 | \( 1 + (2.73 - 2.73i)T - 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 7.16iT - 29T^{2} \) |
| 31 | \( 1 + (-2.46 + 2.46i)T - 31iT^{2} \) |
| 37 | \( 1 + (3.83 + 3.83i)T + 37iT^{2} \) |
| 41 | \( 1 + (-3.97 - 3.97i)T + 41iT^{2} \) |
| 43 | \( 1 - 2.19iT - 43T^{2} \) |
| 47 | \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \) |
| 53 | \( 1 - 0.779iT - 53T^{2} \) |
| 59 | \( 1 + (2.12 - 2.12i)T - 59iT^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + (-4.19 + 4.19i)T - 67iT^{2} \) |
| 71 | \( 1 + (-2.12 - 2.12i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.901 - 0.901i)T + 73iT^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + (2.90 + 2.90i)T + 83iT^{2} \) |
| 89 | \( 1 + (-6.59 + 6.59i)T - 89iT^{2} \) |
| 97 | \( 1 + (-1.19 + 1.19i)T - 97iT^{2} \) |
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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
−10.38944185757482115527260778924, −9.320084431193250199589874343840, −8.743935436451837376555831432215, −8.166219684321302425145448990359, −6.88118478051464889277968739741, −6.01277582964385769881732186966, −4.24048787821216242431788172293, −3.19355799776355652363837878802, −1.86409971757795049891123667437, −0.58545186054593009240629220487,
2.43264035447499391738041254986, 3.65424075026996709634827584185, 4.56750075553390732978128345481, 6.30202954267422248234760012087, 7.06184607916575008035406751528, 7.85268065540074910227855693081, 8.902019208091325792050765311449, 9.231274952455898055982742412943, 10.17229851684325048006872814343, 11.19863105912816197112772435643