| L(s) = 1 | − 3·2-s − 3·3-s + 4-s + 9·6-s − 7·7-s + 21·8-s + 9·9-s + 11·11-s − 3·12-s − 16·13-s + 21·14-s − 71·16-s + 21·17-s − 27·18-s + 125·19-s + 21·21-s − 33·22-s − 81·23-s − 63·24-s + 48·26-s − 27·27-s − 7·28-s + 186·29-s − 58·31-s + 45·32-s − 33·33-s − 63·34-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 1/8·4-s + 0.612·6-s − 0.377·7-s + 0.928·8-s + 1/3·9-s + 0.301·11-s − 0.0721·12-s − 0.341·13-s + 0.400·14-s − 1.10·16-s + 0.299·17-s − 0.353·18-s + 1.50·19-s + 0.218·21-s − 0.319·22-s − 0.734·23-s − 0.535·24-s + 0.362·26-s − 0.192·27-s − 0.0472·28-s + 1.19·29-s − 0.336·31-s + 0.248·32-s − 0.174·33-s − 0.317·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6597021781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6597021781\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + p T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 125 T + p^{3} T^{2} \) |
| 23 | \( 1 + 81 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 58 T + p^{3} T^{2} \) |
| 37 | \( 1 + 253 T + p^{3} T^{2} \) |
| 41 | \( 1 - 63 T + p^{3} T^{2} \) |
| 43 | \( 1 + 100 T + p^{3} T^{2} \) |
| 47 | \( 1 + 219 T + p^{3} T^{2} \) |
| 53 | \( 1 + 192 T + p^{3} T^{2} \) |
| 59 | \( 1 - 249 T + p^{3} T^{2} \) |
| 61 | \( 1 + 64 T + p^{3} T^{2} \) |
| 67 | \( 1 - 272 T + p^{3} T^{2} \) |
| 71 | \( 1 + 645 T + p^{3} T^{2} \) |
| 73 | \( 1 + 112 T + p^{3} T^{2} \) |
| 79 | \( 1 - 509 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1254 T + p^{3} T^{2} \) |
| 89 | \( 1 - 756 T + p^{3} T^{2} \) |
| 97 | \( 1 - 839 T + p^{3} T^{2} \) |
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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
−9.889365764685911901659459268349, −9.140995272377509402815660693960, −8.188000701181918659913910000434, −7.38669700153287403101155319926, −6.58746867955638351422936944006, −5.44764814305653242588115381402, −4.54974943215879527027561154309, −3.29190952807521132315444727282, −1.66144102128139078301751878404, −0.56886215434183115131447864963,
0.56886215434183115131447864963, 1.66144102128139078301751878404, 3.29190952807521132315444727282, 4.54974943215879527027561154309, 5.44764814305653242588115381402, 6.58746867955638351422936944006, 7.38669700153287403101155319926, 8.188000701181918659913910000434, 9.140995272377509402815660693960, 9.889365764685911901659459268349
