| L(s) = 1 | − 27·3-s − 125·5-s + 729·9-s + 3.37e3·15-s − 9.39e3·17-s − 1.31e4·19-s − 1.46e4·23-s + 1.56e4·25-s − 1.96e4·27-s + 5.75e3·31-s − 9.11e4·45-s + 9.00e4·47-s + 1.17e5·49-s + 2.53e5·51-s − 8.86e4·53-s + 3.55e5·57-s − 3.25e5·61-s + 3.95e5·69-s − 4.21e5·75-s + 8.93e5·79-s + 5.31e5·81-s + 4.69e5·83-s + 1.17e6·85-s − 1.55e5·93-s + 1.64e6·95-s − 2.44e6·107-s + 7.73e5·109-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 9-s + 15-s − 1.91·17-s − 1.92·19-s − 1.20·23-s + 25-s − 27-s + 0.193·31-s − 45-s + 0.867·47-s + 49-s + 1.91·51-s − 0.595·53-s + 1.92·57-s − 1.43·61-s + 1.20·69-s − 75-s + 1.81·79-s + 81-s + 0.821·83-s + 1.91·85-s − 0.193·93-s + 1.92·95-s − 1.99·107-s + 0.597·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.4593374649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4593374649\) |
| \(L(4)\) |
|
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 | \( 1 + p^{3} T \) |
| good | 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 + 9394 T + p^{6} T^{2} \) |
| 19 | \( 1 + 13178 T + p^{6} T^{2} \) |
| 23 | \( 1 + 14654 T + p^{6} T^{2} \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 - 5758 T + p^{6} T^{2} \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 90034 T + p^{6} T^{2} \) |
| 53 | \( 1 + 88666 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 + 325798 T + p^{6} T^{2} \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 - 893662 T + p^{6} T^{2} \) |
| 83 | \( 1 - 469546 T + p^{6} T^{2} \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−10.99687310184002560029119318048, −10.54972945291726736644481536933, −9.066957402883309134609795447632, −8.053734064161851765726803437139, −6.88853197818849542369345216128, −6.14089651088027346183299567929, −4.61964481046968679719597776063, −4.03717559707484910111162689304, −2.10871483972589374433328740646, −0.37411264935240343753979239172,
0.37411264935240343753979239172, 2.10871483972589374433328740646, 4.03717559707484910111162689304, 4.61964481046968679719597776063, 6.14089651088027346183299567929, 6.88853197818849542369345216128, 8.053734064161851765726803437139, 9.066957402883309134609795447632, 10.54972945291726736644481536933, 10.99687310184002560029119318048
