| L(s) = 1 | + 6·3-s + 125·5-s − 706·7-s − 2.15e3·9-s + 3.84e3·11-s + 4.05e3·13-s + 750·15-s + 858·17-s − 2.10e4·19-s − 4.23e3·21-s + 8.53e4·23-s + 1.56e4·25-s − 2.60e4·27-s + 8.31e4·29-s − 1.45e5·31-s + 2.30e4·33-s − 8.82e4·35-s + 4.98e5·37-s + 2.43e4·39-s − 6.89e5·41-s − 8.67e5·43-s − 2.68e5·45-s + 2.35e5·47-s − 3.25e5·49-s + 5.14e3·51-s − 1.83e6·53-s + 4.80e5·55-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 0.447·5-s − 0.777·7-s − 0.983·9-s + 0.869·11-s + 0.511·13-s + 0.0573·15-s + 0.0423·17-s − 0.703·19-s − 0.0998·21-s + 1.46·23-s + 1/5·25-s − 0.254·27-s + 0.632·29-s − 0.877·31-s + 0.111·33-s − 0.347·35-s + 1.61·37-s + 0.0656·39-s − 1.56·41-s − 1.66·43-s − 0.439·45-s + 0.331·47-s − 0.394·49-s + 0.00543·51-s − 1.69·53-s + 0.389·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 - 2 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 706 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3840 T + p^{7} T^{2} \) |
| 13 | \( 1 - 4054 T + p^{7} T^{2} \) |
| 17 | \( 1 - 858 T + p^{7} T^{2} \) |
| 19 | \( 1 + 21044 T + p^{7} T^{2} \) |
| 23 | \( 1 - 85338 T + p^{7} T^{2} \) |
| 29 | \( 1 - 83106 T + p^{7} T^{2} \) |
| 31 | \( 1 + 145564 T + p^{7} T^{2} \) |
| 37 | \( 1 - 498886 T + p^{7} T^{2} \) |
| 41 | \( 1 + 689514 T + p^{7} T^{2} \) |
| 43 | \( 1 + 867890 T + p^{7} T^{2} \) |
| 47 | \( 1 - 235638 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1835442 T + p^{7} T^{2} \) |
| 59 | \( 1 + 629508 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2667958 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3373306 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2600052 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1628494 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4243528 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1251378 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6299466 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3976514 T + p^{7} T^{2} \) |
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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
−9.817099975780497485211484427280, −9.037227140738558035715186638277, −8.293718045273612908805765316225, −6.78015517909331211083978041810, −6.21120256770775680539322646221, −5.06300394569654439134028273495, −3.64236185355574158492747346257, −2.75001260268207517163483391121, −1.36030009187276144569128711928, 0,
1.36030009187276144569128711928, 2.75001260268207517163483391121, 3.64236185355574158492747346257, 5.06300394569654439134028273495, 6.21120256770775680539322646221, 6.78015517909331211083978041810, 8.293718045273612908805765316225, 9.037227140738558035715186638277, 9.817099975780497485211484427280
