| L(s) = 1 | + 27·3-s + 64·4-s + 286·7-s + 729·9-s + 1.72e3·12-s − 506·13-s + 4.09e3·16-s − 1.05e4·19-s + 7.72e3·21-s + 1.96e4·27-s + 1.83e4·28-s + 3.52e4·31-s + 4.66e4·36-s + 8.92e4·37-s − 1.36e4·39-s − 1.11e5·43-s + 1.10e5·48-s − 3.58e4·49-s − 3.23e4·52-s − 2.85e5·57-s − 4.20e5·61-s + 2.08e5·63-s + 2.62e5·64-s − 1.72e5·67-s − 6.38e5·73-s − 6.77e5·76-s − 2.04e5·79-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s + 0.833·7-s + 9-s + 12-s − 0.230·13-s + 16-s − 1.54·19-s + 0.833·21-s + 27-s + 0.833·28-s + 1.18·31-s + 36-s + 1.76·37-s − 0.230·39-s − 1.40·43-s + 48-s − 0.304·49-s − 0.230·52-s − 1.54·57-s − 1.85·61-s + 0.833·63-s + 64-s − 0.574·67-s − 1.64·73-s − 1.54·76-s − 0.415·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.551856058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.551856058\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 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
−13.40322268034627153425769892085, −12.23682657755720627213823219175, −11.06764332868845672537737876472, −10.01198749221729798007824601902, −8.504596785111313594136755600501, −7.65191168829179693291548864484, −6.38995870996961307573186034498, −4.45797318955214575603676612236, −2.76797813870064018906311121440, −1.60714911858400301155132080589,
1.60714911858400301155132080589, 2.76797813870064018906311121440, 4.45797318955214575603676612236, 6.38995870996961307573186034498, 7.65191168829179693291548864484, 8.504596785111313594136755600501, 10.01198749221729798007824601902, 11.06764332868845672537737876472, 12.23682657755720627213823219175, 13.40322268034627153425769892085
