| L(s) = 1 | − 2-s − 3·3-s − 7·4-s + 20·5-s + 3·6-s + 2·7-s + 15·8-s + 9·9-s − 20·10-s + 48·11-s + 21·12-s − 14·13-s − 2·14-s − 60·15-s + 41·16-s − 9·18-s + 92·19-s − 140·20-s − 6·21-s − 48·22-s + 122·23-s − 45·24-s + 275·25-s + 14·26-s − 27·27-s − 14·28-s + 36·29-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s + 1.78·5-s + 0.204·6-s + 0.107·7-s + 0.662·8-s + 1/3·9-s − 0.632·10-s + 1.31·11-s + 0.505·12-s − 0.298·13-s − 0.0381·14-s − 1.03·15-s + 0.640·16-s − 0.117·18-s + 1.11·19-s − 1.56·20-s − 0.0623·21-s − 0.465·22-s + 1.10·23-s − 0.382·24-s + 11/5·25-s + 0.105·26-s − 0.192·27-s − 0.0944·28-s + 0.230·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.956312347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956312347\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 122 T + p^{3} T^{2} \) |
| 29 | \( 1 - 36 T + p^{3} T^{2} \) |
| 31 | \( 1 - 182 T + p^{3} T^{2} \) |
| 37 | \( 1 + 76 T + p^{3} T^{2} \) |
| 41 | \( 1 + 294 T + p^{3} T^{2} \) |
| 43 | \( 1 + 428 T + p^{3} T^{2} \) |
| 47 | \( 1 + 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 234 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 820 T + p^{3} T^{2} \) |
| 67 | \( 1 - 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 794 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 79 | \( 1 + 858 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 710 T + p^{3} T^{2} \) |
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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
−9.733315333511607263132046604521, −9.217131978158046630137579897274, −8.356810216087657500687898837456, −6.96506981823629140582068463245, −6.31498195622205121570658352714, −5.25464018597892127253150094746, −4.79699490095752453422319006673, −3.31001550782075612033898086894, −1.69521340929463115826627162072, −0.936073759113514798807751161514,
0.936073759113514798807751161514, 1.69521340929463115826627162072, 3.31001550782075612033898086894, 4.79699490095752453422319006673, 5.25464018597892127253150094746, 6.31498195622205121570658352714, 6.96506981823629140582068463245, 8.356810216087657500687898837456, 9.217131978158046630137579897274, 9.733315333511607263132046604521
