| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 2·9-s − 10-s − 2·13-s + 16-s − 2·18-s − 20-s − 4·25-s − 2·26-s − 29-s + 32-s − 2·36-s − 2·37-s − 40-s + 2·45-s − 10·49-s − 4·50-s − 2·52-s + 18·53-s − 58-s + 16·61-s + 64-s + 2·65-s − 2·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.554·13-s + 1/4·16-s − 0.471·18-s − 0.223·20-s − 4/5·25-s − 0.392·26-s − 0.185·29-s + 0.176·32-s − 1/3·36-s − 0.328·37-s − 0.158·40-s + 0.298·45-s − 1.42·49-s − 0.565·50-s − 0.277·52-s + 2.47·53-s − 0.131·58-s + 2.04·61-s + 1/8·64-s + 0.248·65-s − 0.235·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.074419887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074419887\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−12.24752266560561787626708937144, −11.75724319434331046911995079603, −11.42262378604819619640231114098, −10.71510132489455923711557499190, −10.04790356923855190062379566794, −9.438335836055969991982572410793, −8.552498602810976983008394934086, −8.057302515951047874120594001758, −7.27477984381521693693341581175, −6.69155425315990944048295628417, −5.75917520444999126111897467134, −5.21578319658777085761029440871, −4.25072964409940756727392562206, −3.45278265441740226464914131661, −2.34466519399784613586568724272,
2.34466519399784613586568724272, 3.45278265441740226464914131661, 4.25072964409940756727392562206, 5.21578319658777085761029440871, 5.75917520444999126111897467134, 6.69155425315990944048295628417, 7.27477984381521693693341581175, 8.057302515951047874120594001758, 8.552498602810976983008394934086, 9.438335836055969991982572410793, 10.04790356923855190062379566794, 10.71510132489455923711557499190, 11.42262378604819619640231114098, 11.75724319434331046911995079603, 12.24752266560561787626708937144
