| L(s) = 1 | − 18·3-s − 160·5-s + 343·7-s − 1.86e3·9-s + 5.70e3·11-s − 1.38e3·13-s + 2.88e3·15-s − 3.14e4·17-s − 1.99e4·19-s − 6.17e3·21-s + 7.71e4·23-s − 5.25e4·25-s + 7.29e4·27-s + 1.93e5·29-s + 2.63e4·31-s − 1.02e5·33-s − 5.48e4·35-s − 2.04e5·37-s + 2.49e4·39-s − 6.63e5·41-s − 3.35e5·43-s + 2.98e5·45-s − 1.11e6·47-s + 1.17e5·49-s + 5.65e5·51-s − 1.12e5·53-s − 9.12e5·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.572·5-s + 0.377·7-s − 0.851·9-s + 1.29·11-s − 0.175·13-s + 0.220·15-s − 1.55·17-s − 0.667·19-s − 0.145·21-s + 1.32·23-s − 0.672·25-s + 0.712·27-s + 1.47·29-s + 0.158·31-s − 0.497·33-s − 0.216·35-s − 0.663·37-s + 0.0674·39-s − 1.50·41-s − 0.644·43-s + 0.487·45-s − 1.57·47-s + 1/7·49-s + 0.597·51-s − 0.104·53-s − 0.739·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.162681426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162681426\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 + 2 p^{2} T + p^{7} T^{2} \) |
| 5 | \( 1 + 32 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 5704 T + p^{7} T^{2} \) |
| 13 | \( 1 + 1388 T + p^{7} T^{2} \) |
| 17 | \( 1 + 31434 T + p^{7} T^{2} \) |
| 19 | \( 1 + 19966 T + p^{7} T^{2} \) |
| 23 | \( 1 - 77136 T + p^{7} T^{2} \) |
| 29 | \( 1 - 193374 T + p^{7} T^{2} \) |
| 31 | \( 1 - 26356 T + p^{7} T^{2} \) |
| 37 | \( 1 + 204346 T + p^{7} T^{2} \) |
| 41 | \( 1 + 663050 T + p^{7} T^{2} \) |
| 43 | \( 1 + 335920 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1119812 T + p^{7} T^{2} \) |
| 53 | \( 1 + 112782 T + p^{7} T^{2} \) |
| 59 | \( 1 - 536154 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1170264 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3890660 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2505344 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1435070 T + p^{7} T^{2} \) |
| 79 | \( 1 + 176536 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6211622 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4729062 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2129562 T + p^{7} T^{2} \) |
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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
−9.993890066649116205869225995083, −8.710684404926252805993124090992, −8.451576768950176007142856546120, −6.91643734069742743646818874571, −6.42700258147556906816129720331, −5.07797450898794160174557147706, −4.27974436744002549400129040208, −3.11924247954211185404143053736, −1.79462848292050852304936463167, −0.49002177640491648519700075563,
0.49002177640491648519700075563, 1.79462848292050852304936463167, 3.11924247954211185404143053736, 4.27974436744002549400129040208, 5.07797450898794160174557147706, 6.42700258147556906816129720331, 6.91643734069742743646818874571, 8.451576768950176007142856546120, 8.710684404926252805993124090992, 9.993890066649116205869225995083
