| L(s) = 1 | − 9·9-s + 112·11-s + 8·19-s + 412·29-s + 304·31-s − 492·41-s + 286·49-s + 112·59-s − 4·61-s + 1.34e3·71-s + 816·79-s + 81·81-s − 132·89-s − 1.00e3·99-s − 396·101-s − 124·109-s + 6.74e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 3.00e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.06·11-s + 0.0965·19-s + 2.63·29-s + 1.76·31-s − 1.87·41-s + 0.833·49-s + 0.247·59-s − 0.00839·61-s + 2.24·71-s + 1.16·79-s + 1/9·81-s − 0.157·89-s − 1.02·99-s − 0.390·101-s − 0.108·109-s + 5.06·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.36·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.644931691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.644931691\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 286 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3002 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1410 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 5838 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 21782 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 p T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10730 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 206046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 281878 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 56 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 450982 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 672 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 590866 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 408 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 697350 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 967870 T^{2} + p^{6} T^{4} \) |
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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
−9.406796270532507186065119506631, −9.394674322430231149283179768572, −8.515487653353979054777323147702, −8.484145948769183994606692122498, −8.267303204421392882992577446584, −7.44793473150257063748269153910, −6.76423126404104404986495258869, −6.72059908860130636866462487689, −6.41146539613463882150664044349, −6.00542890556566072510439275248, −5.27220941413690978310297195609, −4.83977581181504672679906650210, −4.16450814229142523975130457252, −4.16426985833152982925715931980, −3.29710813990738109382980153547, −3.09108496034264184932674020248, −2.23080781387141908835824324157, −1.63726890529727013751254769197, −0.900893807151826026147232691003, −0.76757007837128213292428623143,
0.76757007837128213292428623143, 0.900893807151826026147232691003, 1.63726890529727013751254769197, 2.23080781387141908835824324157, 3.09108496034264184932674020248, 3.29710813990738109382980153547, 4.16426985833152982925715931980, 4.16450814229142523975130457252, 4.83977581181504672679906650210, 5.27220941413690978310297195609, 6.00542890556566072510439275248, 6.41146539613463882150664044349, 6.72059908860130636866462487689, 6.76423126404104404986495258869, 7.44793473150257063748269153910, 8.267303204421392882992577446584, 8.484145948769183994606692122498, 8.515487653353979054777323147702, 9.394674322430231149283179768572, 9.406796270532507186065119506631
