| L(s) = 1 | − 4·2-s + 16·4-s + 31·5-s − 230·7-s − 64·8-s − 124·10-s − 121·11-s + 112·13-s + 920·14-s + 256·16-s + 1.14e3·17-s − 612·19-s + 496·20-s + 484·22-s + 1.94e3·23-s − 2.16e3·25-s − 448·26-s − 3.68e3·28-s − 1.19e3·29-s − 1.03e3·31-s − 1.02e3·32-s − 4.56e3·34-s − 7.13e3·35-s + 8.08e3·37-s + 2.44e3·38-s − 1.98e3·40-s + 1.04e4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.554·5-s − 1.77·7-s − 0.353·8-s − 0.392·10-s − 0.301·11-s + 0.183·13-s + 1.25·14-s + 1/4·16-s + 0.958·17-s − 0.388·19-s + 0.277·20-s + 0.213·22-s + 0.765·23-s − 0.692·25-s − 0.129·26-s − 0.887·28-s − 0.263·29-s − 0.193·31-s − 0.176·32-s − 0.677·34-s − 0.983·35-s + 0.970·37-s + 0.275·38-s − 0.196·40-s + 0.970·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.090174784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090174784\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 31 T + p^{5} T^{2} \) |
| 7 | \( 1 + 230 T + p^{5} T^{2} \) |
| 13 | \( 1 - 112 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1142 T + p^{5} T^{2} \) |
| 19 | \( 1 + 612 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1941 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1192 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1037 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8083 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10444 T + p^{5} T^{2} \) |
| 43 | \( 1 - 58 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8656 T + p^{5} T^{2} \) |
| 53 | \( 1 - 20318 T + p^{5} T^{2} \) |
| 59 | \( 1 - 21351 T + p^{5} T^{2} \) |
| 61 | \( 1 - 47044 T + p^{5} T^{2} \) |
| 67 | \( 1 - 48093 T + p^{5} T^{2} \) |
| 71 | \( 1 - 24967 T + p^{5} T^{2} \) |
| 73 | \( 1 + 42288 T + p^{5} T^{2} \) |
| 79 | \( 1 + 72410 T + p^{5} T^{2} \) |
| 83 | \( 1 - 15806 T + p^{5} T^{2} \) |
| 89 | \( 1 - 114761 T + p^{5} T^{2} \) |
| 97 | \( 1 + 5159 T + p^{5} T^{2} \) |
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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
−11.45918511232239833731102974421, −10.20455563811982124882830526216, −9.745485821135642580821225572570, −8.832437047515753055911335137160, −7.50069841812930093766477184119, −6.45963131958783381465462785277, −5.63033844459433588362379537373, −3.60447421163393970094931961810, −2.44950750075559044878740753954, −0.69760500094001640597683267259,
0.69760500094001640597683267259, 2.44950750075559044878740753954, 3.60447421163393970094931961810, 5.63033844459433588362379537373, 6.45963131958783381465462785277, 7.50069841812930093766477184119, 8.832437047515753055911335137160, 9.745485821135642580821225572570, 10.20455563811982124882830526216, 11.45918511232239833731102974421
