| L(s) = 1 | − 2·5-s + 2·9-s + 8·11-s − 8·19-s − 25-s − 4·29-s + 4·41-s − 4·45-s + 10·49-s − 16·55-s − 24·59-s − 20·61-s − 16·71-s + 32·79-s − 5·81-s − 12·89-s + 16·95-s + 16·99-s + 12·101-s + 12·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2/3·9-s + 2.41·11-s − 1.83·19-s − 1/5·25-s − 0.742·29-s + 0.624·41-s − 0.596·45-s + 10/7·49-s − 2.15·55-s − 3.12·59-s − 2.56·61-s − 1.89·71-s + 3.60·79-s − 5/9·81-s − 1.27·89-s + 1.64·95-s + 1.60·99-s + 1.19·101-s + 1.14·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8612374875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8612374875\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−14.87516829368151963944878567818, −14.03027895351106023041294264280, −13.80403053219716984980398795584, −12.91997052424831310620284565891, −12.27661252309399260434141940896, −12.15060959609603369715962295845, −11.46522605854082072550973007185, −10.86241726330800794944868649596, −10.44721562698896132870423004877, −9.352390215483697261062375577735, −9.210189714894171573445856454892, −8.531576274232393632586704936129, −7.67898330238913181870458570066, −7.21583158999792306410434978903, −6.36127294369327897367988140029, −6.06383267017174124142765982170, −4.44387010686589718708577246840, −4.26729765342000317017177688222, −3.43992358269873926659198241096, −1.71634013614640719267961657709,
1.71634013614640719267961657709, 3.43992358269873926659198241096, 4.26729765342000317017177688222, 4.44387010686589718708577246840, 6.06383267017174124142765982170, 6.36127294369327897367988140029, 7.21583158999792306410434978903, 7.67898330238913181870458570066, 8.531576274232393632586704936129, 9.210189714894171573445856454892, 9.352390215483697261062375577735, 10.44721562698896132870423004877, 10.86241726330800794944868649596, 11.46522605854082072550973007185, 12.15060959609603369715962295845, 12.27661252309399260434141940896, 12.91997052424831310620284565891, 13.80403053219716984980398795584, 14.03027895351106023041294264280, 14.87516829368151963944878567818
