| 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
| \(L(3)\) |
\(\approx\) |
\(1.203144122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203144122\) |
| \(L(\frac{7}{2})\) |
|
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
−10.32420311064483840618478082366, −9.273273649944034076682151569830, −8.604477334230504527039486017468, −7.78123591708624660672883285174, −6.44888688624800706063964990948, −5.54599480967236040029572686715, −4.37558308226033282198303751761, −3.64726564178505425956973178917, −2.03328322919460391173021457910, −0.56680113614704972683413546198,
0.56680113614704972683413546198, 2.03328322919460391173021457910, 3.64726564178505425956973178917, 4.37558308226033282198303751761, 5.54599480967236040029572686715, 6.44888688624800706063964990948, 7.78123591708624660672883285174, 8.604477334230504527039486017468, 9.273273649944034076682151569830, 10.32420311064483840618478082366
