| L(s) = 1 | − 27·3-s − 397·7-s + 729·9-s − 4.03e3·13-s − 2.26e3·19-s + 1.07e4·21-s − 1.96e4·27-s − 5.92e4·31-s − 8.92e4·37-s + 1.08e5·39-s + 4.25e4·43-s + 3.99e4·49-s + 6.12e4·57-s + 3.57e5·61-s − 2.89e5·63-s − 5.85e5·67-s + 6.38e5·73-s − 2.04e5·79-s + 5.31e5·81-s + 1.60e6·91-s + 1.59e6·93-s + 1.60e6·97-s + 1.12e6·103-s + 2.30e6·109-s + 2.40e6·111-s − 2.94e6·117-s + ⋯ |
| L(s) = 1 | − 3-s − 1.15·7-s + 9-s − 1.83·13-s − 0.330·19-s + 1.15·21-s − 27-s − 1.98·31-s − 1.76·37-s + 1.83·39-s + 0.535·43-s + 0.339·49-s + 0.330·57-s + 1.57·61-s − 1.15·63-s − 1.94·67-s + 1.64·73-s − 0.415·79-s + 81-s + 2.12·91-s + 1.98·93-s + 1.76·97-s + 1.03·103-s + 1.78·109-s + 1.76·111-s − 1.83·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4276139206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4276139206\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 397 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 + 4033 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 + 2269 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 + 59221 T + p^{6} T^{2} \) |
| 37 | \( 1 + 89206 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 42587 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 - 357839 T + p^{6} T^{2} \) |
| 67 | \( 1 + 585397 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 638066 T + p^{6} T^{2} \) |
| 79 | \( 1 + 204622 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 - 1608263 T + p^{6} T^{2} \) |
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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
−10.60256751608709961316949200289, −9.916789112123746003005278402984, −9.105513501269110809312001948343, −7.42945382911910148115945242965, −6.83589660968384631085331032860, −5.73557443466336455672967011147, −4.82856429340305788199656658907, −3.53894834051771632874106720340, −2.07121372697418947926587764517, −0.33992614383103360464222020122,
0.33992614383103360464222020122, 2.07121372697418947926587764517, 3.53894834051771632874106720340, 4.82856429340305788199656658907, 5.73557443466336455672967011147, 6.83589660968384631085331032860, 7.42945382911910148115945242965, 9.105513501269110809312001948343, 9.916789112123746003005278402984, 10.60256751608709961316949200289
