| L(s) = 1 | + 343·7-s + 729·9-s + 1.96e3·11-s + 2.27e4·23-s + 1.56e4·25-s + 2.12e4·29-s − 1.01e5·37-s − 1.26e5·43-s + 1.17e5·49-s − 5.03e4·53-s + 2.50e5·63-s − 5.39e4·67-s + 2.42e5·71-s + 6.72e5·77-s − 9.29e5·79-s + 5.31e5·81-s + 1.43e6·99-s + 4.63e4·107-s + 2.58e6·109-s − 2.43e6·113-s + ⋯ |
| L(s) = 1 | + 7-s + 9-s + 1.47·11-s + 1.86·23-s + 25-s + 0.870·29-s − 1.99·37-s − 1.59·43-s + 49-s − 0.338·53-s + 63-s − 0.179·67-s + 0.677·71-s + 1.47·77-s − 1.88·79-s + 81-s + 1.47·99-s + 0.0378·107-s + 1.99·109-s − 1.68·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.475358231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.475358231\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( 1 - 1962 T + p^{6} T^{2} \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( 1 - 22734 T + p^{6} T^{2} \) |
| 29 | \( 1 - 21222 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( 1 + 101194 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 + 126614 T + p^{6} T^{2} \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( 1 + 50346 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 + 53926 T + p^{6} T^{2} \) |
| 71 | \( 1 - 242478 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 + 929378 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−10.17350632995548545727074508917, −9.079512125139941468306296833247, −8.458750738991167628950658128523, −7.13950758590313430047248197144, −6.68183491128957916688120962252, −5.13949215897389281862647903919, −4.43826224331333266866007668485, −3.28759531102770418256258428580, −1.67933471948555487527395574046, −1.01765060804365061456293282038,
1.01765060804365061456293282038, 1.67933471948555487527395574046, 3.28759531102770418256258428580, 4.43826224331333266866007668485, 5.13949215897389281862647903919, 6.68183491128957916688120962252, 7.13950758590313430047248197144, 8.458750738991167628950658128523, 9.079512125139941468306296833247, 10.17350632995548545727074508917
