| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 0.885·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 1.77·10-s − 20.7·11-s + 12·12-s + 63.4·13-s + 14·14-s − 2.65·15-s + 16·16-s + 24.8·17-s + 18·18-s + 19·19-s − 3.54·20-s + 21·21-s − 41.5·22-s + 25.1·23-s + 24·24-s − 124.·25-s + 126.·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0791·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.0559·10-s − 0.568·11-s + 0.288·12-s + 1.35·13-s + 0.267·14-s − 0.0457·15-s + 0.250·16-s + 0.354·17-s + 0.235·18-s + 0.229·19-s − 0.0395·20-s + 0.218·21-s − 0.402·22-s + 0.227·23-s + 0.204·24-s − 0.993·25-s + 0.957·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.533351234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.533351234\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 + 0.885T + 125T^{2} \) |
| 11 | \( 1 + 20.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 25.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 96.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 229.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 366.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 103.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 17.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 391.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 559.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 447.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 121.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 383.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 61.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 565.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 46.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 985.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 50.1T + 9.12e5T^{2} \) |
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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
−10.02132100938595711973855008872, −8.855087730016624514720983549979, −8.126649128981266111483843896905, −7.34877648761220098750387712905, −6.25607378738567503035677299484, −5.39580021801657994841405638614, −4.30882985931477329778017932793, −3.45643302920972689105877094511, −2.40777490577310529650148331946, −1.13436664074848958339492832244,
1.13436664074848958339492832244, 2.40777490577310529650148331946, 3.45643302920972689105877094511, 4.30882985931477329778017932793, 5.39580021801657994841405638614, 6.25607378738567503035677299484, 7.34877648761220098750387712905, 8.126649128981266111483843896905, 8.855087730016624514720983549979, 10.02132100938595711973855008872
