| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 7-s − 2·9-s + 4·11-s − 2·12-s + 13-s + 2·14-s − 4·16-s + 2·17-s − 4·18-s + 2·19-s − 21-s + 8·22-s + 3·23-s + 2·26-s + 5·27-s + 2·28-s + 29-s + 2·31-s − 8·32-s − 4·33-s + 4·34-s − 4·36-s + 10·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s − 2/3·9-s + 1.20·11-s − 0.577·12-s + 0.277·13-s + 0.534·14-s − 16-s + 0.485·17-s − 0.942·18-s + 0.458·19-s − 0.218·21-s + 1.70·22-s + 0.625·23-s + 0.392·26-s + 0.962·27-s + 0.377·28-s + 0.185·29-s + 0.359·31-s − 1.41·32-s − 0.696·33-s + 0.685·34-s − 2/3·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.288669985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.288669985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.000129525175821041715307513111, −8.227133394957395871611600424933, −7.07698588628130368434178516310, −6.36779614906058401125503184385, −5.74080072402529902915265408347, −5.07370557695263704041865270035, −4.26714348035994189477569463490, −3.46618061752239067918857276685, −2.54262257780483879586228221565, −1.03032014672593062979582585130,
1.03032014672593062979582585130, 2.54262257780483879586228221565, 3.46618061752239067918857276685, 4.26714348035994189477569463490, 5.07370557695263704041865270035, 5.74080072402529902915265408347, 6.36779614906058401125503184385, 7.07698588628130368434178516310, 8.227133394957395871611600424933, 9.000129525175821041715307513111
