| L(s) = 1 | − 128·4-s + 2.00e3·13-s + 1.63e4·16-s + 1.43e4·19-s − 7.81e4·25-s − 3.31e5·31-s + 3.35e5·37-s − 4.09e5·43-s − 2.57e5·52-s − 3.53e6·61-s − 2.09e6·64-s + 4.44e6·67-s + 1.23e6·73-s − 1.83e6·76-s − 4.51e6·79-s + 1.22e7·97-s + 1.00e7·100-s + 1.38e7·103-s − 9.92e6·109-s + ⋯ |
| L(s) = 1 | − 4-s + 0.253·13-s + 16-s + 0.480·19-s − 25-s − 1.99·31-s + 1.08·37-s − 0.785·43-s − 0.253·52-s − 1.99·61-s − 64-s + 1.80·67-s + 0.372·73-s − 0.480·76-s − 1.03·79-s + 1.36·97-s + 100-s + 1.25·103-s − 0.733·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.250001701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.250001701\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + p^{7} T^{2} \) |
| 5 | \( 1 + p^{7} T^{2} \) |
| 11 | \( 1 + p^{7} T^{2} \) |
| 13 | \( 1 - 2009 T + p^{7} T^{2} \) |
| 17 | \( 1 + p^{7} T^{2} \) |
| 19 | \( 1 - 14357 T + p^{7} T^{2} \) |
| 23 | \( 1 + p^{7} T^{2} \) |
| 29 | \( 1 + p^{7} T^{2} \) |
| 31 | \( 1 + 331387 T + p^{7} T^{2} \) |
| 37 | \( 1 - 335663 T + p^{7} T^{2} \) |
| 41 | \( 1 + p^{7} T^{2} \) |
| 43 | \( 1 + 409495 T + p^{7} T^{2} \) |
| 47 | \( 1 + p^{7} T^{2} \) |
| 53 | \( 1 + p^{7} T^{2} \) |
| 59 | \( 1 + p^{7} T^{2} \) |
| 61 | \( 1 + 3535546 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4443527 T + p^{7} T^{2} \) |
| 71 | \( 1 + p^{7} T^{2} \) |
| 73 | \( 1 - 1236809 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4517617 T + p^{7} T^{2} \) |
| 83 | \( 1 + p^{7} T^{2} \) |
| 89 | \( 1 + p^{7} T^{2} \) |
| 97 | \( 1 - 12245198 T + p^{7} T^{2} \) |
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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
−9.754284980886331210838754193574, −9.165557026401443550923665252862, −8.194452700787984120294584056456, −7.37836247198631311202964292552, −6.03670568172014111248576621128, −5.20503786379960517547541517472, −4.15588793263577434207893285071, −3.28241586064927471653752046612, −1.74762352012486803332080866415, −0.51438705047324064757682398576,
0.51438705047324064757682398576, 1.74762352012486803332080866415, 3.28241586064927471653752046612, 4.15588793263577434207893285071, 5.20503786379960517547541517472, 6.03670568172014111248576621128, 7.37836247198631311202964292552, 8.194452700787984120294584056456, 9.165557026401443550923665252862, 9.754284980886331210838754193574
