| L(s) = 1 | + 2-s + 4-s + 8-s − 2·13-s + 16-s + 9·23-s + 8·25-s − 2·26-s + 32-s − 11·37-s + 9·46-s − 3·47-s + 2·49-s + 8·50-s − 2·52-s + 18·59-s + 7·61-s + 64-s − 3·71-s − 14·73-s − 11·74-s + 15·83-s + 9·92-s − 3·94-s + 13·97-s + 2·98-s + 8·100-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.554·13-s + 1/4·16-s + 1.87·23-s + 8/5·25-s − 0.392·26-s + 0.176·32-s − 1.80·37-s + 1.32·46-s − 0.437·47-s + 2/7·49-s + 1.13·50-s − 0.277·52-s + 2.34·59-s + 0.896·61-s + 1/8·64-s − 0.356·71-s − 1.63·73-s − 1.27·74-s + 1.64·83-s + 0.938·92-s − 0.309·94-s + 1.31·97-s + 0.202·98-s + 4/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.745238414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.745238414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.931780808188515673614912523485, −8.692288678771451087074101239996, −8.132002175622773488255037642468, −7.37996047307283642343094070849, −7.03963848102652834772744146996, −6.77672540888314009976635349024, −6.15957815678936807236897448230, −5.33793866460415731967135369197, −5.13910958292320465610266185922, −4.68381894255141142320285990910, −3.92494938554371250131093377907, −3.28146615247231600325118847487, −2.80811790332849345111107766929, −2.03395041126503383056099309043, −0.982958660413083415809358506184,
0.982958660413083415809358506184, 2.03395041126503383056099309043, 2.80811790332849345111107766929, 3.28146615247231600325118847487, 3.92494938554371250131093377907, 4.68381894255141142320285990910, 5.13910958292320465610266185922, 5.33793866460415731967135369197, 6.15957815678936807236897448230, 6.77672540888314009976635349024, 7.03963848102652834772744146996, 7.37996047307283642343094070849, 8.132002175622773488255037642468, 8.692288678771451087074101239996, 8.931780808188515673614912523485
