| L(s) = 1 | + 0.618·2-s + 3-s − 1.61·4-s + 3.61·5-s + 0.618·6-s + 7-s − 2.23·8-s + 9-s + 2.23·10-s + 11-s − 1.61·12-s + 0.618·13-s + 0.618·14-s + 3.61·15-s + 1.85·16-s − 5.47·17-s + 0.618·18-s + 4.23·19-s − 5.85·20-s + 21-s + 0.618·22-s + 23-s − 2.23·24-s + 8.09·25-s + 0.381·26-s + 27-s − 1.61·28-s + ⋯ |
| L(s) = 1 | + 0.437·2-s + 0.577·3-s − 0.809·4-s + 1.61·5-s + 0.252·6-s + 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.707·10-s + 0.301·11-s − 0.467·12-s + 0.171·13-s + 0.165·14-s + 0.934·15-s + 0.463·16-s − 1.32·17-s + 0.145·18-s + 0.971·19-s − 1.30·20-s + 0.218·21-s + 0.131·22-s + 0.208·23-s − 0.456·24-s + 1.61·25-s + 0.0749·26-s + 0.192·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.292814995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.292814995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 0.618T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 3.85T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.0901T + 53T^{2} \) |
| 59 | \( 1 + 3.61T + 59T^{2} \) |
| 61 | \( 1 + 7.85T + 61T^{2} \) |
| 67 | \( 1 + 8.09T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−10.81527660649945414976973685101, −9.883853675703476396976187449017, −9.102666662268302116038186060727, −8.718857592445549267898173247998, −7.25858215220100463093501573265, −6.09140108542775404097550834575, −5.28527823728134241165798506356, −4.29191789780391757303113677118, −2.94766514301093849313159544074, −1.64314884892201059797517287380,
1.64314884892201059797517287380, 2.94766514301093849313159544074, 4.29191789780391757303113677118, 5.28527823728134241165798506356, 6.09140108542775404097550834575, 7.25858215220100463093501573265, 8.718857592445549267898173247998, 9.102666662268302116038186060727, 9.883853675703476396976187449017, 10.81527660649945414976973685101
