| L(s) = 1 | + 2·4-s − 29·16-s − 80·19-s + 104·31-s + 56·49-s − 88·61-s − 92·64-s − 160·76-s − 56·79-s + 280·109-s + 448·121-s + 208·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 424·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.81·16-s − 4.21·19-s + 3.35·31-s + 8/7·49-s − 1.44·61-s − 1.43·64-s − 2.10·76-s − 0.708·79-s + 2.56·109-s + 3.70·121-s + 1.67·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2.50·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7830896522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7830896522\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2}( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 p T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 224 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 212 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 466 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 1030 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1610 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 1604 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 320 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3194 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3970 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1550 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4784 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6122 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 8290 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7904 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18314 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.823591286066384544964609765432, −8.529595333213368713920655111300, −8.272656334490200088131503145574, −8.025600263840239737755999330761, −7.46360821760587602184042979171, −7.38161315369540023169646876468, −7.20265095047199058899465486400, −6.52617123230284920621000946189, −6.45185028100608453922774452074, −6.31398144062691241059449284850, −6.27548230695885076146624175486, −5.91993959767732348168681032740, −5.24511381819518476455370923540, −4.96735298161233952164142898068, −4.56740235815045195560222265329, −4.41480822564967980399111077866, −4.23739862473275538878131488710, −3.91285342762432973155287660059, −3.31355597429707771266312885729, −2.78912513467632958469077964012, −2.40392633734657348304349419978, −2.31901649433285768253414893739, −1.88075454730649538828532941852, −1.14804305772938842096207146812, −0.24345738659617943704166198109,
0.24345738659617943704166198109, 1.14804305772938842096207146812, 1.88075454730649538828532941852, 2.31901649433285768253414893739, 2.40392633734657348304349419978, 2.78912513467632958469077964012, 3.31355597429707771266312885729, 3.91285342762432973155287660059, 4.23739862473275538878131488710, 4.41480822564967980399111077866, 4.56740235815045195560222265329, 4.96735298161233952164142898068, 5.24511381819518476455370923540, 5.91993959767732348168681032740, 6.27548230695885076146624175486, 6.31398144062691241059449284850, 6.45185028100608453922774452074, 6.52617123230284920621000946189, 7.20265095047199058899465486400, 7.38161315369540023169646876468, 7.46360821760587602184042979171, 8.025600263840239737755999330761, 8.272656334490200088131503145574, 8.529595333213368713920655111300, 8.823591286066384544964609765432
