| L(s) = 1 | − 4-s − 9-s + 2·11-s + 16-s − 20·29-s − 16·31-s + 36-s + 4·41-s − 2·44-s + 10·49-s − 16·61-s − 64-s + 4·71-s − 20·79-s + 81-s − 20·89-s − 2·99-s + 4·101-s − 40·109-s + 20·116-s + 3·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 0.603·11-s + 1/4·16-s − 3.71·29-s − 2.87·31-s + 1/6·36-s + 0.624·41-s − 0.301·44-s + 10/7·49-s − 2.04·61-s − 1/8·64-s + 0.474·71-s − 2.25·79-s + 1/9·81-s − 2.11·89-s − 0.201·99-s + 0.398·101-s − 3.83·109-s + 1.85·116-s + 3/11·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7073005888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7073005888\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| 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
−9.451244925899781260341753514716, −9.123463998767731192261584686087, −9.070591803761554420789480223931, −8.404486179123643248992889284048, −8.030692359906575778140429624992, −7.40173628750019723564117687706, −7.29470672646151145759988212956, −7.00279095886970434614985254435, −6.20153241684398480419995395848, −5.77153859670218542435856309852, −5.44891750653020413083088438986, −5.36736935519956745607453607461, −4.41770100531740954584044121436, −4.04187603849126722323463787764, −3.78449737398514082784193751406, −3.25124176518631853222042880894, −2.62890812677439565918998337133, −1.68356166782088212191799807964, −1.68290142557913494877747392579, −0.31662098383637041834441383576,
0.31662098383637041834441383576, 1.68290142557913494877747392579, 1.68356166782088212191799807964, 2.62890812677439565918998337133, 3.25124176518631853222042880894, 3.78449737398514082784193751406, 4.04187603849126722323463787764, 4.41770100531740954584044121436, 5.36736935519956745607453607461, 5.44891750653020413083088438986, 5.77153859670218542435856309852, 6.20153241684398480419995395848, 7.00279095886970434614985254435, 7.29470672646151145759988212956, 7.40173628750019723564117687706, 8.030692359906575778140429624992, 8.404486179123643248992889284048, 9.070591803761554420789480223931, 9.123463998767731192261584686087, 9.451244925899781260341753514716
