| L(s) = 1 | + 8·2-s + 64·4-s + 512·8-s + 2.33e3·11-s + 4.09e3·16-s + 1.72e3·17-s − 2.48e3·19-s + 1.87e4·22-s + 1.56e4·25-s + 3.27e4·32-s + 1.38e4·34-s − 1.98e4·38-s − 1.34e5·41-s − 7.49e4·43-s + 1.49e5·44-s + 1.17e5·49-s + 1.25e5·50-s − 3.04e5·59-s + 2.62e5·64-s − 5.96e5·67-s + 1.10e5·68-s − 5.93e5·73-s − 1.58e5·76-s − 1.07e6·82-s − 6.78e5·83-s − 5.99e5·86-s + 1.19e6·88-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 8-s + 1.75·11-s + 16-s + 0.351·17-s − 0.361·19-s + 1.75·22-s + 25-s + 32-s + 0.351·34-s − 0.361·38-s − 1.95·41-s − 0.942·43-s + 1.75·44-s + 49-s + 50-s − 1.48·59-s + 64-s − 1.98·67-s + 0.351·68-s − 1.52·73-s − 0.361·76-s − 1.95·82-s − 1.18·83-s − 0.942·86-s + 1.75·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.870536198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.870536198\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( 1 - 2338 T + p^{6} T^{2} \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 - 1726 T + p^{6} T^{2} \) |
| 19 | \( 1 + 2482 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 + 134642 T + p^{6} T^{2} \) |
| 43 | \( 1 + 74914 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( 1 + 304958 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 + 596626 T + p^{6} T^{2} \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 + 593134 T + p^{6} T^{2} \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( 1 + 678926 T + p^{6} T^{2} \) |
| 89 | \( 1 - 357262 T + p^{6} T^{2} \) |
| 97 | \( 1 - 1822754 T + p^{6} T^{2} \) |
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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
−13.51225828642595810649751988690, −12.28901075524337143111726006031, −11.55988509987007774099999632055, −10.28520872202215563412857764446, −8.780661444958387118734222466214, −7.12305919479374606977053946719, −6.13381903371298534855228064032, −4.60878756639366435991665836769, −3.34516930045231986892629436143, −1.49751559914752858477471144769,
1.49751559914752858477471144769, 3.34516930045231986892629436143, 4.60878756639366435991665836769, 6.13381903371298534855228064032, 7.12305919479374606977053946719, 8.780661444958387118734222466214, 10.28520872202215563412857764446, 11.55988509987007774099999632055, 12.28901075524337143111726006031, 13.51225828642595810649751988690
