| L(s) = 1 | − 5.05·2-s + 3·3-s + 17.5·4-s − 5·5-s − 15.1·6-s + 19.3·7-s − 48.0·8-s + 9·9-s + 25.2·10-s + 6.81·11-s + 52.5·12-s + 36.9·13-s − 97.9·14-s − 15·15-s + 102.·16-s − 71.5·17-s − 45.4·18-s + 88.3·19-s − 87.5·20-s + 58.1·21-s − 34.4·22-s + 185.·23-s − 144.·24-s + 25·25-s − 186.·26-s + 27·27-s + 339.·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.18·4-s − 0.447·5-s − 1.03·6-s + 1.04·7-s − 2.12·8-s + 0.333·9-s + 0.798·10-s + 0.186·11-s + 1.26·12-s + 0.788·13-s − 1.86·14-s − 0.258·15-s + 1.60·16-s − 1.02·17-s − 0.595·18-s + 1.06·19-s − 0.978·20-s + 0.604·21-s − 0.333·22-s + 1.68·23-s − 1.22·24-s + 0.200·25-s − 1.40·26-s + 0.192·27-s + 2.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.140032156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140032156\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 5.05T + 8T^{2} \) |
| 7 | \( 1 - 19.3T + 343T^{2} \) |
| 11 | \( 1 - 6.81T + 1.33e3T^{2} \) |
| 13 | \( 1 - 36.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 185.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 66.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 42.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 9.42T + 1.48e5T^{2} \) |
| 59 | \( 1 + 232.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 546.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 953.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 429.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 554.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 965.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 625.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 271.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−10.80383635026862864231431039112, −9.490839594659093023199702024390, −8.849194633638707096754678496914, −8.207983135785276456917364210385, −7.42161245175559202207243362990, −6.62690730911304084259681656298, −4.94029591447359055997234220526, −3.35009682765527178709407456479, −1.94887964615063909224104847769, −0.912823117500316736515447719411,
0.912823117500316736515447719411, 1.94887964615063909224104847769, 3.35009682765527178709407456479, 4.94029591447359055997234220526, 6.62690730911304084259681656298, 7.42161245175559202207243362990, 8.207983135785276456917364210385, 8.849194633638707096754678496914, 9.490839594659093023199702024390, 10.80383635026862864231431039112
