| L(s) = 1 | + 243·3-s + 1.87e3·5-s − 7.23e4·7-s + 5.90e4·9-s + 1.47e5·11-s − 1.56e6·13-s + 4.54e5·15-s − 1.45e5·17-s + 1.09e6·19-s − 1.75e7·21-s − 6.00e7·23-s − 4.53e7·25-s + 1.43e7·27-s − 1.96e7·29-s − 2.39e8·31-s + 3.59e7·33-s − 1.35e8·35-s + 4.88e8·37-s − 3.79e8·39-s + 4.70e7·41-s + 4.28e8·43-s + 1.10e8·45-s + 4.50e8·47-s + 3.25e9·49-s − 3.54e7·51-s + 4.33e9·53-s + 2.76e8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.267·5-s − 1.62·7-s + 1/3·9-s + 0.276·11-s − 1.16·13-s + 0.154·15-s − 0.0249·17-s + 0.101·19-s − 0.938·21-s − 1.94·23-s − 0.928·25-s + 0.192·27-s − 0.177·29-s − 1.50·31-s + 0.159·33-s − 0.435·35-s + 1.15·37-s − 0.674·39-s + 0.0634·41-s + 0.444·43-s + 0.0892·45-s + 0.286·47-s + 1.64·49-s − 0.0143·51-s + 1.42·53-s + 0.0741·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 - 374 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 72312 T + p^{11} T^{2} \) |
| 11 | \( 1 - 147940 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1562858 T + p^{11} T^{2} \) |
| 17 | \( 1 + 145774 T + p^{11} T^{2} \) |
| 19 | \( 1 - 1096796 T + p^{11} T^{2} \) |
| 23 | \( 1 + 60014264 T + p^{11} T^{2} \) |
| 29 | \( 1 + 19626954 T + p^{11} T^{2} \) |
| 31 | \( 1 + 239950480 T + p^{11} T^{2} \) |
| 37 | \( 1 - 488238078 T + p^{11} T^{2} \) |
| 41 | \( 1 - 47066010 T + p^{11} T^{2} \) |
| 43 | \( 1 - 428866948 T + p^{11} T^{2} \) |
| 47 | \( 1 - 450903216 T + p^{11} T^{2} \) |
| 53 | \( 1 - 4336685950 T + p^{11} T^{2} \) |
| 59 | \( 1 + 8937556460 T + p^{11} T^{2} \) |
| 61 | \( 1 - 4673884486 T + p^{11} T^{2} \) |
| 67 | \( 1 - 7498937612 T + p^{11} T^{2} \) |
| 71 | \( 1 + 27032101480 T + p^{11} T^{2} \) |
| 73 | \( 1 - 159947146 p T + p^{11} T^{2} \) |
| 79 | \( 1 - 2478876544 T + p^{11} T^{2} \) |
| 83 | \( 1 - 42745596956 T + p^{11} T^{2} \) |
| 89 | \( 1 + 93270772662 T + p^{11} T^{2} \) |
| 97 | \( 1 - 118032786914 T + p^{11} T^{2} \) |
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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
−14.41297998522453337062311051850, −13.22843922359218547184044647190, −12.15786874775221019944580020966, −10.05839546974392453051291703944, −9.315567654893807959216058477644, −7.47574620600309386098192087867, −6.02178808990392592474071709002, −3.82055112567291176513883686895, −2.32731464365634502448968700085, 0,
2.32731464365634502448968700085, 3.82055112567291176513883686895, 6.02178808990392592474071709002, 7.47574620600309386098192087867, 9.315567654893807959216058477644, 10.05839546974392453051291703944, 12.15786874775221019944580020966, 13.22843922359218547184044647190, 14.41297998522453337062311051850
