| L(s) = 1 | − 4-s − 7-s + 2·9-s − 4·11-s − 4·13-s + 16-s + 4·17-s − 5·25-s + 28-s − 6·29-s − 2·36-s + 6·37-s − 8·43-s + 4·44-s − 4·47-s − 4·49-s + 4·52-s + 6·53-s − 14·59-s − 2·63-s − 64-s − 4·68-s + 4·77-s − 5·81-s − 20·89-s + 4·91-s − 8·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.377·7-s + 2/3·9-s − 1.20·11-s − 1.10·13-s + 1/4·16-s + 0.970·17-s − 25-s + 0.188·28-s − 1.11·29-s − 1/3·36-s + 0.986·37-s − 1.21·43-s + 0.603·44-s − 0.583·47-s − 4/7·49-s + 0.554·52-s + 0.824·53-s − 1.82·59-s − 0.251·63-s − 1/8·64-s − 0.485·68-s + 0.455·77-s − 5/9·81-s − 2.11·89-s + 0.419·91-s − 0.804·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78652 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78652 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.596622664474202731592788570117, −9.229273132476331003248850070653, −8.346782881308694755112328895326, −7.963153156794202053416906364592, −7.52070638365227441812313209417, −7.13934541618123167462108986216, −6.37214490288566521623571174666, −5.68406294894437403218413429182, −5.29087851095645022515248904215, −4.68367372710157577032037359800, −4.07729785464170928063882398282, −3.30891607901095725965242961863, −2.64609489416475809292646958474, −1.64895781968600437415584718959, 0,
1.64895781968600437415584718959, 2.64609489416475809292646958474, 3.30891607901095725965242961863, 4.07729785464170928063882398282, 4.68367372710157577032037359800, 5.29087851095645022515248904215, 5.68406294894437403218413429182, 6.37214490288566521623571174666, 7.13934541618123167462108986216, 7.52070638365227441812313209417, 7.963153156794202053416906364592, 8.346782881308694755112328895326, 9.229273132476331003248850070653, 9.596622664474202731592788570117
