| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s − 7-s − 2·9-s − 2·10-s − 5·11-s − 2·12-s + 3·13-s − 2·14-s + 15-s − 4·16-s − 5·17-s − 4·18-s − 2·20-s + 21-s − 10·22-s − 23-s + 25-s + 6·26-s + 5·27-s − 2·28-s + 3·29-s + 2·30-s + 6·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s − 2/3·9-s − 0.632·10-s − 1.50·11-s − 0.577·12-s + 0.832·13-s − 0.534·14-s + 0.258·15-s − 16-s − 1.21·17-s − 0.942·18-s − 0.447·20-s + 0.218·21-s − 2.13·22-s − 0.208·23-s + 1/5·25-s + 1.17·26-s + 0.962·27-s − 0.377·28-s + 0.557·29-s + 0.365·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−10.19088948384714538493823702201, −8.815775547612089210365847118991, −8.088997236407093398136755587737, −6.73583674342883967453753798377, −6.14576399565931630270892532621, −5.22615496619836725047595629293, −4.56794494570433530257579526757, −3.39517376220725744927159548680, −2.55266460829949316145212153910, 0,
2.55266460829949316145212153910, 3.39517376220725744927159548680, 4.56794494570433530257579526757, 5.22615496619836725047595629293, 6.14576399565931630270892532621, 6.73583674342883967453753798377, 8.088997236407093398136755587737, 8.815775547612089210365847118991, 10.19088948384714538493823702201
