| L(s) = 1 | + 8·3-s − 10·5-s + 49·7-s − 179·9-s − 340·11-s + 294·13-s − 80·15-s + 1.22e3·17-s + 2.43e3·19-s + 392·21-s − 2.00e3·23-s − 3.02e3·25-s − 3.37e3·27-s + 6.74e3·29-s − 8.85e3·31-s − 2.72e3·33-s − 490·35-s − 9.18e3·37-s + 2.35e3·39-s − 1.45e4·41-s + 8.10e3·43-s + 1.79e3·45-s + 312·47-s + 2.40e3·49-s + 9.80e3·51-s + 1.46e4·53-s + 3.40e3·55-s + ⋯ |
| L(s) = 1 | + 0.513·3-s − 0.178·5-s + 0.377·7-s − 0.736·9-s − 0.847·11-s + 0.482·13-s − 0.0918·15-s + 1.02·17-s + 1.54·19-s + 0.193·21-s − 0.788·23-s − 0.967·25-s − 0.891·27-s + 1.48·29-s − 1.65·31-s − 0.434·33-s − 0.0676·35-s − 1.10·37-s + 0.247·39-s − 1.35·41-s + 0.668·43-s + 0.131·45-s + 0.0206·47-s + 1/7·49-s + 0.528·51-s + 0.715·53-s + 0.151·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 8 T + p^{5} T^{2} \) |
| 5 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 - 294 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1226 T + p^{5} T^{2} \) |
| 19 | \( 1 - 128 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 2000 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6746 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8856 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14574 T + p^{5} T^{2} \) |
| 43 | \( 1 - 8108 T + p^{5} T^{2} \) |
| 47 | \( 1 - 312 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14634 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27656 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34338 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12316 T + p^{5} T^{2} \) |
| 71 | \( 1 + 520 p T + p^{5} T^{2} \) |
| 73 | \( 1 + 61718 T + p^{5} T^{2} \) |
| 79 | \( 1 - 64752 T + p^{5} T^{2} \) |
| 83 | \( 1 + 77056 T + p^{5} T^{2} \) |
| 89 | \( 1 + 8166 T + p^{5} T^{2} \) |
| 97 | \( 1 - 20650 T + p^{5} T^{2} \) |
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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
−9.839408870992683160379570553760, −8.810502527670251520167186835465, −7.998813648732990376034057146470, −7.39766495080756663760798384201, −5.85919048402179928673300543283, −5.18042622679993216672667935714, −3.69458849090888699630004513230, −2.86594637199296251226259995622, −1.52598710338013181656378090409, 0,
1.52598710338013181656378090409, 2.86594637199296251226259995622, 3.69458849090888699630004513230, 5.18042622679993216672667935714, 5.85919048402179928673300543283, 7.39766495080756663760798384201, 7.998813648732990376034057146470, 8.810502527670251520167186835465, 9.839408870992683160379570553760
