| L(s) = 1 | + 4-s − 3·5-s − 3·11-s + 16-s + 4·19-s − 3·20-s + 4·25-s + 6·29-s + 31-s + 12·41-s − 3·44-s + 5·49-s + 9·55-s + 12·59-s − 2·61-s + 64-s + 18·71-s + 4·76-s − 2·79-s − 3·80-s − 12·95-s + 4·100-s − 21·101-s + 22·109-s + 6·116-s − 13·121-s + 124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s − 0.904·11-s + 1/4·16-s + 0.917·19-s − 0.670·20-s + 4/5·25-s + 1.11·29-s + 0.179·31-s + 1.87·41-s − 0.452·44-s + 5/7·49-s + 1.21·55-s + 1.56·59-s − 0.256·61-s + 1/8·64-s + 2.13·71-s + 0.458·76-s − 0.225·79-s − 0.335·80-s − 1.23·95-s + 2/5·100-s − 2.08·101-s + 2.10·109-s + 0.557·116-s − 1.18·121-s + 0.0898·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.182148566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.182148566\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 83 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−9.756066611170383196558631242057, −9.430447163480861087117723152963, −8.514759819180443725312676267069, −8.318792364444255930653538442194, −7.65944456346811281849398343724, −7.43805688912822900313636661018, −6.86289527264364971472073148731, −6.23405456241996150252281646948, −5.52016494898208972860392207075, −5.01935493779612780932617074438, −4.27826378492955433676349042871, −3.72829403587239693691099622825, −2.96250672757091686807822027871, −2.37693242111623456644600156491, −0.870291249680570153072547881963,
0.870291249680570153072547881963, 2.37693242111623456644600156491, 2.96250672757091686807822027871, 3.72829403587239693691099622825, 4.27826378492955433676349042871, 5.01935493779612780932617074438, 5.52016494898208972860392207075, 6.23405456241996150252281646948, 6.86289527264364971472073148731, 7.43805688912822900313636661018, 7.65944456346811281849398343724, 8.318792364444255930653538442194, 8.514759819180443725312676267069, 9.430447163480861087117723152963, 9.756066611170383196558631242057
