| L(s) = 1 | + 2.85·2-s + 3·3-s + 0.154·4-s − 2.88·5-s + 8.56·6-s + 19.3·7-s − 22.4·8-s + 9·9-s − 8.23·10-s + 45.0·11-s + 0.464·12-s + 92.2·13-s + 55.3·14-s − 8.65·15-s − 65.2·16-s + 4.16·17-s + 25.7·18-s + 92.2·19-s − 0.446·20-s + 58.1·21-s + 128.·22-s − 178.·23-s − 67.2·24-s − 116.·25-s + 263.·26-s + 27·27-s + 2.99·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.577·3-s + 0.0193·4-s − 0.257·5-s + 0.582·6-s + 1.04·7-s − 0.990·8-s + 0.333·9-s − 0.260·10-s + 1.23·11-s + 0.0111·12-s + 1.96·13-s + 1.05·14-s − 0.148·15-s − 1.01·16-s + 0.0594·17-s + 0.336·18-s + 1.11·19-s − 0.00498·20-s + 0.603·21-s + 1.24·22-s − 1.61·23-s − 0.571·24-s − 0.933·25-s + 1.98·26-s + 0.192·27-s + 0.0202·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.550858146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.550858146\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 - 67T \) |
| good | 2 | \( 1 - 2.85T + 8T^{2} \) |
| 5 | \( 1 + 2.88T + 125T^{2} \) |
| 7 | \( 1 - 19.3T + 343T^{2} \) |
| 11 | \( 1 - 45.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.16T + 4.91e3T^{2} \) |
| 19 | \( 1 - 92.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 63.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 368.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 318.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 176.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 900.T + 2.26e5T^{2} \) |
| 71 | \( 1 + 991.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 178.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 437.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 899.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 534.T + 9.12e5T^{2} \) |
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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
−11.79163053584246636815635226486, −11.64905541560337212074036991744, −10.00803223166182029570782310951, −8.741773231241397903821384993303, −8.185805590447077498374243811593, −6.60214353385501044211509660798, −5.48959779412331360941476729965, −4.13068248902769447424992056522, −3.53022705484398343647026273617, −1.51336905830804213118484200295,
1.51336905830804213118484200295, 3.53022705484398343647026273617, 4.13068248902769447424992056522, 5.48959779412331360941476729965, 6.60214353385501044211509660798, 8.185805590447077498374243811593, 8.741773231241397903821384993303, 10.00803223166182029570782310951, 11.64905541560337212074036991744, 11.79163053584246636815635226486
