| L(s) = 1 | − 2.14·2-s + 6.84·3-s − 3.38·4-s − 17.1·5-s − 14.7·6-s + 22.9·7-s + 24.4·8-s + 19.9·9-s + 36.8·10-s − 49.3·11-s − 23.1·12-s + 15.4·13-s − 49.3·14-s − 117.·15-s − 25.4·16-s + 71.6·17-s − 42.7·18-s − 59.6·19-s + 58.0·20-s + 157.·21-s + 105.·22-s + 161.·23-s + 167.·24-s + 168.·25-s − 33.1·26-s − 48.5·27-s − 77.8·28-s + ⋯ |
| L(s) = 1 | − 0.759·2-s + 1.31·3-s − 0.423·4-s − 1.53·5-s − 1.00·6-s + 1.24·7-s + 1.08·8-s + 0.737·9-s + 1.16·10-s − 1.35·11-s − 0.558·12-s + 0.329·13-s − 0.942·14-s − 2.02·15-s − 0.397·16-s + 1.02·17-s − 0.559·18-s − 0.720·19-s + 0.648·20-s + 1.63·21-s + 1.02·22-s + 1.46·23-s + 1.42·24-s + 1.34·25-s − 0.249·26-s − 0.346·27-s − 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.376223798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376223798\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.14T + 8T^{2} \) |
| 3 | \( 1 - 6.84T + 27T^{2} \) |
| 5 | \( 1 + 17.1T + 125T^{2} \) |
| 7 | \( 1 - 22.9T + 343T^{2} \) |
| 11 | \( 1 + 49.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.84T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 383.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 532.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 253.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 690.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 636.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 475.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 146.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 783.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 102.T + 9.12e5T^{2} \) |
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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
−8.551454360293835856162213733935, −8.254645021542334428671405956624, −7.61747098355189363314639503214, −7.34464090272227243745712004561, −5.35682206830265274715371962870, −4.62051463228255743328821568073, −3.80504162323790007893974639328, −2.97257608783465118137639171498, −1.75949474414354663646171602371, −0.58983717256422655862055226499,
0.58983717256422655862055226499, 1.75949474414354663646171602371, 2.97257608783465118137639171498, 3.80504162323790007893974639328, 4.62051463228255743328821568073, 5.35682206830265274715371962870, 7.34464090272227243745712004561, 7.61747098355189363314639503214, 8.254645021542334428671405956624, 8.551454360293835856162213733935
