| L(s) = 1 | + 4-s − 2·9-s − 11-s + 16-s − 16·23-s + 10·25-s − 2·36-s + 4·37-s − 44-s − 7·49-s − 28·53-s + 64-s − 8·67-s − 5·81-s − 16·92-s + 2·99-s + 10·100-s + 4·113-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 4·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2/3·9-s − 0.301·11-s + 1/4·16-s − 3.33·23-s + 2·25-s − 1/3·36-s + 0.657·37-s − 0.150·44-s − 49-s − 3.84·53-s + 1/8·64-s − 0.977·67-s − 5/9·81-s − 1.66·92-s + 0.201·99-s + 100-s + 0.376·113-s + 1/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.536757738003163472773938294826, −8.166730852936611115960554871947, −7.82021979681689371707293119014, −7.40789779682054330247393673786, −6.56345269675560905791803261211, −6.33341735807205803870658227629, −5.92221856819849171166113388153, −5.32646135494688323743441817683, −4.66180794577029847515972105997, −4.25135230852627933522009234979, −3.35485886243871762322388210983, −2.97182172523047703774195232195, −2.21072101718741916868872670441, −1.50989331984010194201557058291, 0,
1.50989331984010194201557058291, 2.21072101718741916868872670441, 2.97182172523047703774195232195, 3.35485886243871762322388210983, 4.25135230852627933522009234979, 4.66180794577029847515972105997, 5.32646135494688323743441817683, 5.92221856819849171166113388153, 6.33341735807205803870658227629, 6.56345269675560905791803261211, 7.40789779682054330247393673786, 7.82021979681689371707293119014, 8.166730852936611115960554871947, 8.536757738003163472773938294826
