L(s) = 1 | + 27·3-s + 286·7-s + 729·9-s + 506·13-s + 1.05e4·19-s + 7.72e3·21-s + 1.56e4·25-s + 1.96e4·27-s − 3.52e4·31-s − 8.92e4·37-s + 1.36e4·39-s − 1.11e5·43-s − 3.58e4·49-s + 2.85e5·57-s − 4.20e5·61-s + 2.08e5·63-s − 1.72e5·67-s + 6.38e5·73-s + 4.21e5·75-s + 2.04e5·79-s + 5.31e5·81-s + 1.44e5·91-s − 9.52e5·93-s − 5.64e4·97-s − 1.12e6·103-s − 2.17e6·109-s − 2.40e6·111-s + ⋯ |
L(s) = 1 | + 3-s + 0.833·7-s + 9-s + 0.230·13-s + 1.54·19-s + 0.833·21-s + 25-s + 27-s − 1.18·31-s − 1.76·37-s + 0.230·39-s − 1.40·43-s − 0.304·49-s + 1.54·57-s − 1.85·61-s + 0.833·63-s − 0.574·67-s + 1.64·73-s + 75-s + 0.415·79-s + 81-s + 0.192·91-s − 1.18·93-s − 0.0618·97-s − 1.03·103-s − 1.67·109-s − 1.76·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.704291011\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.704291011\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
good | 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( 1 - 286 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 - 506 T + p^{6} T^{2} \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( 1 - 10582 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 + 35282 T + p^{6} T^{2} \) |
| 37 | \( 1 + 89206 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 + 111386 T + p^{6} T^{2} \) |
| 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 | \( 1 + 420838 T + p^{6} T^{2} \) |
| 67 | \( 1 + 172874 T + p^{6} T^{2} \) |
| 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 | \( 1 + 56446 T + p^{6} T^{2} \) |
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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
−14.36036711079201857277652032879, −13.52389063339429235838213761192, −12.16751131503417358404365251918, −10.76564764192639791637036462445, −9.384106802006606075954561307721, −8.270326014845728187440228638173, −7.13427526460697103122579961395, −5.02468006308217887647701318983, −3.33759648271553528484131004379, −1.55590919304050924656819873866,
1.55590919304050924656819873866, 3.33759648271553528484131004379, 5.02468006308217887647701318983, 7.13427526460697103122579961395, 8.270326014845728187440228638173, 9.384106802006606075954561307721, 10.76564764192639791637036462445, 12.16751131503417358404365251918, 13.52389063339429235838213761192, 14.36036711079201857277652032879