| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.00·4-s + 2.23i·5-s + 2.44·6-s − 5.52i·7-s − 2.82·8-s + 2.99·9-s − 3.16i·10-s − 3.99i·11-s − 3.46·12-s + 8.18·13-s + 7.81i·14-s − 3.87i·15-s + 4.00·16-s + 17.3i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.447i·5-s + 0.408·6-s − 0.789i·7-s − 0.353·8-s + 0.333·9-s − 0.316i·10-s − 0.363i·11-s − 0.288·12-s + 0.629·13-s + 0.558i·14-s − 0.258i·15-s + 0.250·16-s + 1.02i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6855221208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6855221208\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (22.7 + 3.11i)T \) |
| good | 7 | \( 1 + 5.52iT - 49T^{2} \) |
| 11 | \( 1 + 3.99iT - 121T^{2} \) |
| 13 | \( 1 - 8.18T + 169T^{2} \) |
| 17 | \( 1 - 17.3iT - 289T^{2} \) |
| 19 | \( 1 + 11.0iT - 361T^{2} \) |
| 29 | \( 1 - 10.3T + 841T^{2} \) |
| 31 | \( 1 + 20.6T + 961T^{2} \) |
| 37 | \( 1 + 39.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 73.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 52.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 74.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 18.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 6.77T + 3.48e3T^{2} \) |
| 61 | \( 1 + 115. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 30.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 89.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 141. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 66.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 54.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 8.89iT - 9.40e3T^{2} \) |
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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
−10.16758501041092267587355728081, −9.291047334682664686241738070533, −8.198405916716207272463887149796, −7.47737132130850622412472856044, −6.46525807345768441756089222527, −5.89638740626332874150850403128, −4.39932216330615887603197859323, −3.35871193997169000003956097688, −1.76080636411348264330578283477, −0.37194185236309882170280513909,
1.20146110839447914743611823168, 2.50202274909120223915067816199, 4.05112157487361118759017510492, 5.31399202755941675908695458009, 6.00802963417444702701638793915, 7.04606344516919938539837700555, 8.016714085442868814539600599674, 8.844168625377800611953506779876, 9.622947676972952578313235568534, 10.36992681280742918910720442252
