| L(s) = 1 | − 3.50·2-s + (−2.88 − 0.822i)3-s + 8.29·4-s + (10.1 + 2.88i)6-s + 2.64i·7-s − 15.0·8-s + (7.64 + 4.74i)9-s + 7.01i·11-s + (−23.9 − 6.82i)12-s + 11.6i·13-s − 9.27i·14-s + 19.5·16-s + 4.52·17-s + (−26.8 − 16.6i)18-s − 16.2·19-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + (−0.961 − 0.274i)3-s + 2.07·4-s + (1.68 + 0.480i)6-s + 0.377i·7-s − 1.88·8-s + (0.849 + 0.527i)9-s + 0.637i·11-s + (−1.99 − 0.568i)12-s + 0.895i·13-s − 0.662i·14-s + 1.22·16-s + 0.266·17-s + (−1.48 − 0.924i)18-s − 0.854·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2509021794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2509021794\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.88 + 0.822i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 2 | \( 1 + 3.50T + 4T^{2} \) |
| 11 | \( 1 - 7.01iT - 121T^{2} \) |
| 13 | \( 1 - 11.6iT - 169T^{2} \) |
| 17 | \( 1 - 4.52T + 289T^{2} \) |
| 19 | \( 1 + 16.2T + 361T^{2} \) |
| 23 | \( 1 - 25.5T + 529T^{2} \) |
| 29 | \( 1 - 9.49iT - 841T^{2} \) |
| 31 | \( 1 - 28.7T + 961T^{2} \) |
| 37 | \( 1 + 33.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 67.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 33.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 15.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 92.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 57.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 15.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 76.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 74.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 23.1iT - 9.40e3T^{2} \) |
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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
−10.79248929326776951025585014227, −10.11609451384325214113471507073, −9.233564641713618833138052718106, −8.472454276649004073935737831114, −7.30181138622171752916101680263, −6.84359794821081223035319738105, −5.81472945787175058487773011994, −4.47797966368383683688004744060, −2.35594460415009739122921186981, −1.26644693876941901087365725537,
0.23989718834349543539577315248, 1.28365855846023892076999212243, 3.09363940805499963968038577497, 4.75569676792971331114875772475, 6.07325040311523508925414051615, 6.76933123819680012363827147818, 7.81290359052971347423357053952, 8.554603133307914506661868828484, 9.639303347457660926720929161704, 10.24653333164115489065411647573
