| L(s) = 1 | + 3.87·2-s − 3.41·3-s + 7.02·4-s + 12.7·5-s − 13.2·6-s − 1.75·7-s − 3.78·8-s − 15.3·9-s + 49.5·10-s − 50.7·11-s − 23.9·12-s + 17.6·13-s − 6.78·14-s − 43.6·15-s − 70.8·16-s + 12.5·17-s − 59.5·18-s − 153.·19-s + 89.8·20-s + 5.97·21-s − 196.·22-s + 12.9·24-s + 38.6·25-s + 68.5·26-s + 144.·27-s − 12.2·28-s + 181.·29-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 0.656·3-s + 0.877·4-s + 1.14·5-s − 0.900·6-s − 0.0945·7-s − 0.167·8-s − 0.568·9-s + 1.56·10-s − 1.39·11-s − 0.576·12-s + 0.377·13-s − 0.129·14-s − 0.751·15-s − 1.10·16-s + 0.179·17-s − 0.779·18-s − 1.85·19-s + 1.00·20-s + 0.0620·21-s − 1.90·22-s + 0.110·24-s + 0.309·25-s + 0.517·26-s + 1.03·27-s − 0.0829·28-s + 1.16·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 3.87T + 8T^{2} \) |
| 3 | \( 1 + 3.41T + 27T^{2} \) |
| 5 | \( 1 - 12.7T + 125T^{2} \) |
| 7 | \( 1 + 1.75T + 343T^{2} \) |
| 11 | \( 1 + 50.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 153.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 87.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 50.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 408.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 73.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 189.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 4.68T + 3.00e5T^{2} \) |
| 71 | \( 1 + 867.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 553.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 606.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 823.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 381.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−10.42851138550993977260967436858, −9.170618552655585061884246105920, −8.182288399362096347625307755180, −6.58318831841944626271156029336, −6.02153461492518425142811412210, −5.34101891475077198904699778706, −4.55444899911359151676715893871, −3.08241318782938946871161411230, −2.14330113912419483348279021267, 0,
2.14330113912419483348279021267, 3.08241318782938946871161411230, 4.55444899911359151676715893871, 5.34101891475077198904699778706, 6.02153461492518425142811412210, 6.58318831841944626271156029336, 8.182288399362096347625307755180, 9.170618552655585061884246105920, 10.42851138550993977260967436858
