| L(s) = 1 | + 5.65i·2-s − 15.9·4-s − 29.6i·5-s − 73.1·7-s + 0.427i·8-s + 167.·10-s + 200. i·11-s + 240.·13-s − 413. i·14-s − 257.·16-s + 296. i·17-s − 180.·19-s + 471. i·20-s − 1.13e3·22-s − 767. i·23-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 0.995·4-s − 1.18i·5-s − 1.49·7-s + 0.00668i·8-s + 1.67·10-s + 1.65i·11-s + 1.42·13-s − 2.11i·14-s − 1.00·16-s + 1.02i·17-s − 0.499·19-s + 1.17i·20-s − 2.34·22-s − 1.45i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5134106629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5134106629\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 - 5.65iT - 16T^{2} \) |
| 5 | \( 1 + 29.6iT - 625T^{2} \) |
| 7 | \( 1 + 73.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 200. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 240.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 296. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 180.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 767. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.23e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.52e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.52e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.55e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 793.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.96e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.91e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 2.64e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 3.64e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 1.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 473.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.99e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 619. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.20e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.02e3T + 8.85e7T^{2} \) |
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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
−9.840210908860117944320969606088, −8.803287031461121519936288834249, −8.553626572549330400812219214374, −7.19271067438146014397564701293, −6.55218708448256174106970826055, −5.73729661191775149287836865514, −4.71465971964783158203032234151, −3.75566905588164148969399251481, −1.85936460085066341684093704511, −0.14914607802634997048807697074,
1.03361178101121451238378842921, 2.72309472682771408325337329637, 3.23709526108104999889488562093, 3.89094526404399997857198355537, 5.99238431027273775653193906945, 6.39912173659965596705044572203, 7.64815714463661282911007344922, 9.111486086498158041854292610926, 9.572443358047377076421762525778, 10.65916749835441112887279269877
