| L(s) = 1 | − 4.11·2-s − 3·3-s + 8.91·4-s − 20.0·5-s + 12.3·6-s − 5.82·7-s − 3.77·8-s + 9·9-s + 82.4·10-s + 19.8·11-s − 26.7·12-s − 62.3·13-s + 23.9·14-s + 60.1·15-s − 55.8·16-s − 26.4·17-s − 37.0·18-s − 178.·20-s + 17.4·21-s − 81.6·22-s + 90.5·23-s + 11.3·24-s + 276.·25-s + 256.·26-s − 27·27-s − 51.9·28-s + 45.6·29-s + ⋯ |
| L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.11·4-s − 1.79·5-s + 0.839·6-s − 0.314·7-s − 0.166·8-s + 0.333·9-s + 2.60·10-s + 0.544·11-s − 0.643·12-s − 1.32·13-s + 0.457·14-s + 1.03·15-s − 0.872·16-s − 0.377·17-s − 0.484·18-s − 1.99·20-s + 0.181·21-s − 0.791·22-s + 0.821·23-s + 0.0963·24-s + 2.21·25-s + 1.93·26-s − 0.192·27-s − 0.350·28-s + 0.292·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.008114451553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008114451553\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 4.11T + 8T^{2} \) |
| 5 | \( 1 + 20.0T + 125T^{2} \) |
| 7 | \( 1 + 5.82T + 343T^{2} \) |
| 11 | \( 1 - 19.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 90.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 45.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 320.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 75.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 128.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 304.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 250.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 837.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 660.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 103.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 767.T + 9.12e5T^{2} \) |
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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
−9.383576138848858955156643214004, −8.707532856060646584753187661895, −7.85070152988162750513506757828, −7.15201941928107423002264231780, −6.75779978285086447060750905625, −5.07324943283347377754870089951, −4.25963177756549447515640349592, −3.11822897177272457576986935577, −1.50222075737105220652765275870, −0.06254842411839057360129616847,
0.06254842411839057360129616847, 1.50222075737105220652765275870, 3.11822897177272457576986935577, 4.25963177756549447515640349592, 5.07324943283347377754870089951, 6.75779978285086447060750905625, 7.15201941928107423002264231780, 7.85070152988162750513506757828, 8.707532856060646584753187661895, 9.383576138848858955156643214004
