| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 2·5-s + 2·6-s + 9-s − 4·10-s − 5·11-s + 2·12-s − 13-s − 2·15-s − 4·16-s − 17-s + 2·18-s − 8·19-s − 4·20-s − 10·22-s + 6·23-s − 25-s − 2·26-s + 27-s − 8·29-s − 4·30-s − 9·31-s − 8·32-s − 5·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s + 1/3·9-s − 1.26·10-s − 1.50·11-s + 0.577·12-s − 0.277·13-s − 0.516·15-s − 16-s − 0.242·17-s + 0.471·18-s − 1.83·19-s − 0.894·20-s − 2.13·22-s + 1.25·23-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.48·29-s − 0.730·30-s − 1.61·31-s − 1.41·32-s − 0.870·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32487 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913031497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913031497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.99864717883754, −14.77389165650935, −13.89154316901981, −13.30948339087926, −13.01412488303200, −12.66186099059889, −12.12660638226240, −11.37427455322271, −10.81627726658330, −10.66197858305168, −9.555180448692692, −9.069082418259557, −8.438107465073191, −7.838280146300428, −7.326778965580426, −6.825330454696880, −5.986256131596635, −5.412454303858755, −4.847357327141144, −4.237890654173443, −3.761875540347305, −3.173973762920422, −2.397367540103456, −2.031930675516063, −0.3649211691153178,
0.3649211691153178, 2.031930675516063, 2.397367540103456, 3.173973762920422, 3.761875540347305, 4.237890654173443, 4.847357327141144, 5.412454303858755, 5.986256131596635, 6.825330454696880, 7.326778965580426, 7.838280146300428, 8.438107465073191, 9.069082418259557, 9.555180448692692, 10.66197858305168, 10.81627726658330, 11.37427455322271, 12.12660638226240, 12.66186099059889, 13.01412488303200, 13.30948339087926, 13.89154316901981, 14.77389165650935, 14.99864717883754
