| L(s) = 1 | + 4·5-s − 9-s + 4·11-s + 11·25-s + 16·31-s + 4·41-s − 4·45-s + 10·49-s + 16·55-s − 20·59-s − 4·61-s − 24·71-s + 81-s + 20·89-s − 4·99-s + 16·101-s + 20·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·155-s + 157-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s + 1.20·11-s + 11/5·25-s + 2.87·31-s + 0.624·41-s − 0.596·45-s + 10/7·49-s + 2.15·55-s − 2.60·59-s − 0.512·61-s − 2.84·71-s + 1/9·81-s + 2.11·89-s − 0.402·99-s + 1.59·101-s + 1.91·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.14·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.521563073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.521563073\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18364502159992335238648205135, −9.833776562646347466676882747811, −9.384064569249551478990180713425, −8.971978897568705580423844699306, −8.826894411114671847087748345650, −8.322158042952626455600158208771, −7.55950415037675494715112703776, −7.37690217813721676929007350833, −6.54196454830521372494982997938, −6.26610215086726361520580870028, −6.12787929568860191645301468827, −5.66263154031096757822641733587, −4.87625642194862086874035110455, −4.67011774542556332856712106536, −4.08235065221894392875316020330, −3.24016238096554676569682538509, −2.79037509052512667748577503659, −2.25673511107964657709754938345, −1.50590960483716422198531697519, −0.975660376270329757854206180532,
0.975660376270329757854206180532, 1.50590960483716422198531697519, 2.25673511107964657709754938345, 2.79037509052512667748577503659, 3.24016238096554676569682538509, 4.08235065221894392875316020330, 4.67011774542556332856712106536, 4.87625642194862086874035110455, 5.66263154031096757822641733587, 6.12787929568860191645301468827, 6.26610215086726361520580870028, 6.54196454830521372494982997938, 7.37690217813721676929007350833, 7.55950415037675494715112703776, 8.322158042952626455600158208771, 8.826894411114671847087748345650, 8.971978897568705580423844699306, 9.384064569249551478990180713425, 9.833776562646347466676882747811, 10.18364502159992335238648205135
