| L(s) = 1 | − 9·3-s + 26·5-s + 49·7-s + 81·9-s − 664·11-s + 318·13-s − 234·15-s + 1.58e3·17-s − 236·19-s − 441·21-s − 2.21e3·23-s − 2.44e3·25-s − 729·27-s − 4.95e3·29-s + 7.12e3·31-s + 5.97e3·33-s + 1.27e3·35-s + 4.35e3·37-s − 2.86e3·39-s + 1.05e4·41-s + 8.45e3·43-s + 2.10e3·45-s − 5.35e3·47-s + 2.40e3·49-s − 1.42e4·51-s − 3.33e4·53-s − 1.72e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.465·5-s + 0.377·7-s + 1/3·9-s − 1.65·11-s + 0.521·13-s − 0.268·15-s + 1.32·17-s − 0.149·19-s − 0.218·21-s − 0.871·23-s − 0.783·25-s − 0.192·27-s − 1.09·29-s + 1.33·31-s + 0.955·33-s + 0.175·35-s + 0.523·37-s − 0.301·39-s + 0.979·41-s + 0.697·43-s + 0.155·45-s − 0.353·47-s + 1/7·49-s − 0.766·51-s − 1.63·53-s − 0.769·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 - 26 T + p^{5} T^{2} \) |
| 11 | \( 1 + 664 T + p^{5} T^{2} \) |
| 13 | \( 1 - 318 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 19 | \( 1 + 236 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2212 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4954 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7128 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4358 T + p^{5} T^{2} \) |
| 41 | \( 1 - 10542 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8452 T + p^{5} T^{2} \) |
| 47 | \( 1 + 5352 T + p^{5} T^{2} \) |
| 53 | \( 1 + 33354 T + p^{5} T^{2} \) |
| 59 | \( 1 - 15436 T + p^{5} T^{2} \) |
| 61 | \( 1 + 36762 T + p^{5} T^{2} \) |
| 67 | \( 1 + 40972 T + p^{5} T^{2} \) |
| 71 | \( 1 - 9092 T + p^{5} T^{2} \) |
| 73 | \( 1 + 73454 T + p^{5} T^{2} \) |
| 79 | \( 1 + 89400 T + p^{5} T^{2} \) |
| 83 | \( 1 - 6428 T + p^{5} T^{2} \) |
| 89 | \( 1 + 122658 T + p^{5} T^{2} \) |
| 97 | \( 1 - 21370 T + p^{5} T^{2} \) |
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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
−10.30897399029227035244068700272, −9.595064828236385494532302376364, −8.126188003395187090778589637598, −7.56513965107433831794246185639, −6.01203705959371648391091733460, −5.52283042850781084261895152055, −4.32673120490332448328288671949, −2.80209931397049631438706348526, −1.46522290721873986289364986319, 0,
1.46522290721873986289364986319, 2.80209931397049631438706348526, 4.32673120490332448328288671949, 5.52283042850781084261895152055, 6.01203705959371648391091733460, 7.56513965107433831794246185639, 8.126188003395187090778589637598, 9.595064828236385494532302376364, 10.30897399029227035244068700272
