| L(s) = 1 | + 2·3-s − 4-s + 4·5-s − 2·9-s − 4·11-s − 2·12-s + 8·15-s − 3·16-s − 4·20-s + 6·23-s + 2·25-s − 10·27-s + 2·31-s − 8·33-s + 2·36-s − 4·37-s + 4·44-s − 8·45-s + 4·47-s − 6·48-s + 10·49-s − 4·53-s − 16·55-s − 10·59-s − 8·60-s + 7·64-s + 24·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 1.78·5-s − 2/3·9-s − 1.20·11-s − 0.577·12-s + 2.06·15-s − 3/4·16-s − 0.894·20-s + 1.25·23-s + 2/5·25-s − 1.92·27-s + 0.359·31-s − 1.39·33-s + 1/3·36-s − 0.657·37-s + 0.603·44-s − 1.19·45-s + 0.583·47-s − 0.866·48-s + 10/7·49-s − 0.549·53-s − 2.15·55-s − 1.30·59-s − 1.03·60-s + 7/8·64-s + 2.93·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.431509771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431509771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−11.24266824857667382095114905233, −10.84774011024589057102109795171, −10.10560965227450825602482185078, −9.645000873631015039671320140222, −9.176924144304341056325339296270, −8.757424348895816692226973769871, −8.236493774681509535168944957746, −7.57155468986634107877468395952, −6.72692821975313412101796792831, −5.89397677289112679601593035659, −5.44241788196571385030338265716, −4.81688724026962305815801552814, −3.56092321497703829049205121547, −2.60993985257061879919699988566, −2.18833033243502336622929241230,
2.18833033243502336622929241230, 2.60993985257061879919699988566, 3.56092321497703829049205121547, 4.81688724026962305815801552814, 5.44241788196571385030338265716, 5.89397677289112679601593035659, 6.72692821975313412101796792831, 7.57155468986634107877468395952, 8.236493774681509535168944957746, 8.757424348895816692226973769871, 9.176924144304341056325339296270, 9.645000873631015039671320140222, 10.10560965227450825602482185078, 10.84774011024589057102109795171, 11.24266824857667382095114905233
