| L(s) = 1 | + 2-s + 4-s + 8-s − 5·9-s + 4·11-s − 2·13-s + 16-s + 6·17-s − 5·18-s − 19-s + 4·22-s + 6·25-s − 2·26-s − 10·29-s − 16·31-s + 32-s + 6·34-s − 5·36-s − 4·37-s − 38-s + 8·43-s + 4·44-s − 5·49-s + 6·50-s − 2·52-s − 2·53-s − 10·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 1.17·18-s − 0.229·19-s + 0.852·22-s + 6/5·25-s − 0.392·26-s − 1.85·29-s − 2.87·31-s + 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.657·37-s − 0.162·38-s + 1.21·43-s + 0.603·44-s − 5/7·49-s + 0.848·50-s − 0.277·52-s − 0.274·53-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.931225006811838608284657047407, −7.26713629185078190541270557233, −7.25038064772684706053554512263, −6.57921390943921996981302294081, −6.00340543493637960120759847028, −5.67117215547378839234241338271, −5.28069074518443602737935578596, −5.01844065191313724446625296272, −3.93518538194328508890956081033, −3.86057015395193111701613139713, −3.22085281099588136489203399087, −2.76277276227245620478208829006, −1.99718990943742281838117673850, −1.32950412893204015190093848332, 0,
1.32950412893204015190093848332, 1.99718990943742281838117673850, 2.76277276227245620478208829006, 3.22085281099588136489203399087, 3.86057015395193111701613139713, 3.93518538194328508890956081033, 5.01844065191313724446625296272, 5.28069074518443602737935578596, 5.67117215547378839234241338271, 6.00340543493637960120759847028, 6.57921390943921996981302294081, 7.25038064772684706053554512263, 7.26713629185078190541270557233, 7.931225006811838608284657047407
