| L(s) = 1 | − 2.23·2-s − 2·3-s + 3.00·4-s − 0.618·5-s + 4.47·6-s − 3.23·7-s − 2.23·8-s + 9-s + 1.38·10-s + 0.763·11-s − 6.00·12-s − 4.85·13-s + 7.23·14-s + 1.23·15-s − 0.999·16-s − 1.61·17-s − 2.23·18-s − 1.85·20-s + 6.47·21-s − 1.70·22-s + 4.47·23-s + 4.47·24-s − 4.61·25-s + 10.8·26-s + 4·27-s − 9.70·28-s + 4.09·29-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 1.15·3-s + 1.50·4-s − 0.276·5-s + 1.82·6-s − 1.22·7-s − 0.790·8-s + 0.333·9-s + 0.437·10-s + 0.230·11-s − 1.73·12-s − 1.34·13-s + 1.93·14-s + 0.319·15-s − 0.249·16-s − 0.392·17-s − 0.527·18-s − 0.414·20-s + 1.41·21-s − 0.364·22-s + 0.932·23-s + 0.912·24-s − 0.923·25-s + 2.12·26-s + 0.769·27-s − 1.83·28-s + 0.759·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2205501946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2205501946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 4.85T + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 3.14T + 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 71 | \( 1 + 1.23T + 71T^{2} \) |
| 73 | \( 1 - 5.56T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 + 0.0901T + 97T^{2} \) |
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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
−11.19931962696425782819749655061, −10.36867859074742180344313591596, −9.670870665915046097084934038365, −8.902209761802464606038571101334, −7.63063318154922331124185563353, −6.83066724883614345154670562108, −6.03212104130555659448655465473, −4.59929929672457824214516035692, −2.68394029336039325924371764473, −0.58545577887490055659980554211,
0.58545577887490055659980554211, 2.68394029336039325924371764473, 4.59929929672457824214516035692, 6.03212104130555659448655465473, 6.83066724883614345154670562108, 7.63063318154922331124185563353, 8.902209761802464606038571101334, 9.670870665915046097084934038365, 10.36867859074742180344313591596, 11.19931962696425782819749655061
