| L(s) = 1 | − 4.35·2-s − 1.21·3-s + 10.9·4-s + 6.90·5-s + 5.29·6-s − 2.81·7-s − 12.9·8-s − 25.5·9-s − 30.0·10-s + 61.2·11-s − 13.3·12-s − 10.5·13-s + 12.2·14-s − 8.39·15-s − 31.3·16-s − 104.·17-s + 111.·18-s + 70.5·19-s + 75.7·20-s + 3.42·21-s − 266.·22-s + 15.7·24-s − 77.2·25-s + 45.7·26-s + 63.8·27-s − 30.8·28-s − 124.·29-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.233·3-s + 1.37·4-s + 0.617·5-s + 0.360·6-s − 0.152·7-s − 0.572·8-s − 0.945·9-s − 0.951·10-s + 1.67·11-s − 0.320·12-s − 0.224·13-s + 0.234·14-s − 0.144·15-s − 0.490·16-s − 1.48·17-s + 1.45·18-s + 0.851·19-s + 0.847·20-s + 0.0355·21-s − 2.58·22-s + 0.133·24-s − 0.618·25-s + 0.345·26-s + 0.454·27-s − 0.208·28-s − 0.794·29-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 | \( 1 + 4.35T + 8T^{2} \) |
| 3 | \( 1 + 1.21T + 27T^{2} \) |
| 5 | \( 1 - 6.90T + 125T^{2} \) |
| 7 | \( 1 + 2.81T + 343T^{2} \) |
| 11 | \( 1 - 61.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70.5T + 6.85e3T^{2} \) |
| 29 | \( 1 + 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 20.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 377.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 667.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 74.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 65.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 654.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 602.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 116.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 311.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 832.T + 9.12e5T^{2} \) |
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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
−9.501602478617932416283920870387, −9.355601689152847783545434817644, −8.527148768859670547533727140604, −7.37738562811984010908535845713, −6.53607817026640384689312524565, −5.70067758932720217677718215507, −4.13618418925978789432823647400, −2.47931005585333403897019756256, −1.34843714149504822221224644500, 0,
1.34843714149504822221224644500, 2.47931005585333403897019756256, 4.13618418925978789432823647400, 5.70067758932720217677718215507, 6.53607817026640384689312524565, 7.37738562811984010908535845713, 8.527148768859670547533727140604, 9.355601689152847783545434817644, 9.501602478617932416283920870387
