| L(s) = 1 | + 4·2-s − 15·3-s − 16·4-s − 19·5-s − 60·6-s − 10·7-s − 192·8-s − 18·9-s − 76·10-s + 240·12-s + 1.14e3·13-s − 40·14-s + 285·15-s − 256·16-s − 686·17-s − 72·18-s + 384·19-s + 304·20-s + 150·21-s + 3.70e3·23-s + 2.88e3·24-s − 2.76e3·25-s + 4.59e3·26-s + 3.91e3·27-s + 160·28-s + 5.42e3·29-s + 1.14e3·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.962·3-s − 1/2·4-s − 0.339·5-s − 0.680·6-s − 0.0771·7-s − 1.06·8-s − 0.0740·9-s − 0.240·10-s + 0.481·12-s + 1.88·13-s − 0.0545·14-s + 0.327·15-s − 1/4·16-s − 0.575·17-s − 0.0523·18-s + 0.244·19-s + 0.169·20-s + 0.0742·21-s + 1.46·23-s + 1.02·24-s − 0.884·25-s + 1.33·26-s + 1.03·27-s + 0.0385·28-s + 1.19·29-s + 0.231·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.264941339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264941339\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 3 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 19 T + p^{5} T^{2} \) |
| 7 | \( 1 + 10 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1148 T + p^{5} T^{2} \) |
| 17 | \( 1 + 686 T + p^{5} T^{2} \) |
| 19 | \( 1 - 384 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3709 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5424 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6443 T + p^{5} T^{2} \) |
| 37 | \( 1 - 12063 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1528 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4026 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7168 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29862 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6461 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16980 T + p^{5} T^{2} \) |
| 67 | \( 1 - 29999 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31023 T + p^{5} T^{2} \) |
| 73 | \( 1 + 1924 T + p^{5} T^{2} \) |
| 79 | \( 1 + 65138 T + p^{5} T^{2} \) |
| 83 | \( 1 - 102714 T + p^{5} T^{2} \) |
| 89 | \( 1 - 17415 T + p^{5} T^{2} \) |
| 97 | \( 1 - 66905 T + p^{5} T^{2} \) |
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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
−12.63879041091691841271865540068, −11.46506290015585117347447880725, −10.92731482771973942934849217901, −9.292915387969130485317709339640, −8.297502645634824596417323039163, −6.50331079120193744055727719021, −5.68476484897072429485486303605, −4.53475555779189861203758707271, −3.31740389842518171416597768036, −0.74129788734137256761143189721,
0.74129788734137256761143189721, 3.31740389842518171416597768036, 4.53475555779189861203758707271, 5.68476484897072429485486303605, 6.50331079120193744055727719021, 8.297502645634824596417323039163, 9.292915387969130485317709339640, 10.92731482771973942934849217901, 11.46506290015585117347447880725, 12.63879041091691841271865540068
