| L(s) = 1 | + 4·7-s + 8·17-s − 16·23-s + 6·25-s + 12·31-s − 24·41-s + 16·47-s − 2·49-s − 4·73-s + 28·79-s + 16·89-s − 4·97-s + 28·103-s + 16·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 64·161-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.94·17-s − 3.33·23-s + 6/5·25-s + 2.15·31-s − 3.74·41-s + 2.33·47-s − 2/7·49-s − 0.468·73-s + 3.15·79-s + 1.69·89-s − 0.406·97-s + 2.75·103-s + 1.50·113-s + 2.93·119-s + 6/11·121-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 + 0.0798·157-s − 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.240402687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.240402687\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.048025329264663163203931877171, −8.709300795005357538530617332393, −8.210686078121232995263350437880, −8.118423603161609883839111496015, −7.80878656003157104201747380952, −7.48819817400860423132417322315, −6.81756142485846972362362592486, −6.51279934646055292306682525608, −5.88915702127857121684363438003, −5.82855567052060597326322994946, −5.03874594779638127219045257809, −4.91273421180633331399394313953, −4.53405321468064621006556008122, −3.88505626414893136792979567633, −3.46945635230909159625872193192, −3.09662908174040347702916850974, −2.16111727700154012284912434614, −1.98793125147703548963332987721, −1.30197178502154322975593198905, −0.66169946246110650114389106974,
0.66169946246110650114389106974, 1.30197178502154322975593198905, 1.98793125147703548963332987721, 2.16111727700154012284912434614, 3.09662908174040347702916850974, 3.46945635230909159625872193192, 3.88505626414893136792979567633, 4.53405321468064621006556008122, 4.91273421180633331399394313953, 5.03874594779638127219045257809, 5.82855567052060597326322994946, 5.88915702127857121684363438003, 6.51279934646055292306682525608, 6.81756142485846972362362592486, 7.48819817400860423132417322315, 7.80878656003157104201747380952, 8.118423603161609883839111496015, 8.210686078121232995263350437880, 8.709300795005357538530617332393, 9.048025329264663163203931877171
