L(s) = 1 | + 1.80i·2-s − 1.24·4-s + 1.44i·5-s + 3.44i·7-s + 1.35i·8-s − 2.60·10-s + 5.18i·11-s − 6.20·14-s − 4.93·16-s − 0.753·17-s − 7.96i·19-s − 1.80i·20-s − 9.34·22-s − 2.82·23-s + 2.91·25-s + ⋯ |
L(s) = 1 | + 1.27i·2-s − 0.623·4-s + 0.646i·5-s + 1.30i·7-s + 0.479i·8-s − 0.823·10-s + 1.56i·11-s − 1.65·14-s − 1.23·16-s − 0.182·17-s − 1.82i·19-s − 0.402i·20-s − 1.99·22-s − 0.589·23-s + 0.582·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.486184839\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.486184839\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.80iT - 2T^{2} \) |
| 5 | \( 1 - 1.44iT - 5T^{2} \) |
| 7 | \( 1 - 3.44iT - 7T^{2} \) |
| 11 | \( 1 - 5.18iT - 11T^{2} \) |
| 17 | \( 1 + 0.753T + 17T^{2} \) |
| 19 | \( 1 + 7.96iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 3.91T + 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 6.24iT - 37T^{2} \) |
| 41 | \( 1 - 1.80iT - 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 1.87iT - 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 - 4.54iT - 67T^{2} \) |
| 71 | \( 1 + 9.11iT - 71T^{2} \) |
| 73 | \( 1 - 2.95iT - 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 + 6.46iT - 83T^{2} \) |
| 89 | \( 1 + 1.15iT - 89T^{2} \) |
| 97 | \( 1 + 8.65iT - 97T^{2} \) |
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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
−9.719518800965350709827961697482, −8.891516930621033074976266428196, −8.360037282203744467655217899920, −7.17820637206252390613818742613, −6.91579725744595885070731744920, −6.09034477069619678630006253042, −5.11552001072888826541112247010, −4.57736799700435064376539972789, −2.82893593531154431674806716284, −2.15442141077925096724105773819,
0.60450165489209901482015975115, 1.39156330829601708732277765830, 2.77797267102332078703257956137, 3.86436461674460819072309450638, 4.17505597147181952646903374011, 5.59365795538674494871959080542, 6.42807918164682483294406279863, 7.56020525780053532922267981278, 8.299552324280312360657011499180, 9.177245606605690370806338181339