| L(s) = 1 | − 32·4-s − 427·13-s + 1.02e3·16-s + 3.14e3·19-s − 3.12e3·25-s + 2.72e3·31-s − 6.66e3·37-s + 2.24e4·43-s + 1.36e4·52-s − 3.86e4·61-s − 3.27e4·64-s − 3.79e4·67-s − 7.81e4·73-s − 1.00e5·76-s + 9.08e4·79-s − 1.34e5·97-s + 1.00e5·100-s − 2.11e5·103-s − 2.47e5·109-s + ⋯ |
| L(s) = 1 | − 4-s − 0.700·13-s + 16-s + 1.99·19-s − 25-s + 0.508·31-s − 0.799·37-s + 1.85·43-s + 0.700·52-s − 1.32·61-s − 64-s − 1.03·67-s − 1.71·73-s − 1.99·76-s + 1.63·79-s − 1.45·97-s + 100-s − 1.96·103-s − 1.99·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 + 427 T + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 - 3143 T + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 - 2723 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6661 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 - 22475 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + 38626 T + p^{5} T^{2} \) |
| 67 | \( 1 + 37939 T + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 + 78127 T + p^{5} T^{2} \) |
| 79 | \( 1 - 90857 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + 134386 T + p^{5} T^{2} \) |
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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
−9.666372817930952278439595845199, −9.181236535826443319853735024663, −7.991431705689398423379777811938, −7.30992954340692533157741271439, −5.83904394427893921751868609224, −5.04721348512054281481552600152, −4.02392078430049503535293210105, −2.89475856359783892808840131559, −1.23870311755812014528042780686, 0,
1.23870311755812014528042780686, 2.89475856359783892808840131559, 4.02392078430049503535293210105, 5.04721348512054281481552600152, 5.83904394427893921751868609224, 7.30992954340692533157741271439, 7.991431705689398423379777811938, 9.181236535826443319853735024663, 9.666372817930952278439595845199
