| L(s) = 1 | − 4·3-s + 10·9-s + 16·25-s − 20·27-s − 16·29-s + 16·31-s − 16·47-s − 16·53-s − 64·75-s + 35·81-s − 48·83-s + 64·87-s − 64·93-s − 48·103-s + 32·109-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 64·141-s + 149-s + 151-s + 157-s + 64·159-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s + 16/5·25-s − 3.84·27-s − 2.97·29-s + 2.87·31-s − 2.33·47-s − 2.19·53-s − 7.39·75-s + 35/9·81-s − 5.26·83-s + 6.86·87-s − 6.63·93-s − 4.72·103-s + 3.06·109-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.38·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 5.07·159-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1045989007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1045989007\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2:C_4$ | \( 1 - 16 T^{2} + 112 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 48 T^{2} + 1152 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 60 T^{2} + 1830 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 72 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 112 T^{2} + 6400 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 108 T^{2} + 6102 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 192 T^{2} + 16080 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 52 T^{2} + 9142 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 4 T^{2} - 282 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 224 T^{2} + 23104 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 4 T^{2} + 11974 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 144 T^{2} + 20448 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.32647727843282047794853010092, −6.30841562512505362466084337100, −6.03538927915580055116666514504, −5.71520306341466831760459369912, −5.52463584195135138291247305371, −5.32343922505680147297635481476, −5.20873921394203099062030772232, −4.87967084877586384608896834488, −4.76040216370376423020191417908, −4.57977351298623338823676199041, −4.52431060928792218274667896972, −4.08237623411166074745723648960, −3.86233774080046789716250612311, −3.74671946656088102014515150209, −3.29489312991416412464782763490, −3.08725971063358345708231687965, −2.82379802445324777454984750692, −2.52464494927919617870615438511, −2.48449953552213712499924520481, −1.61049140143573995827314163094, −1.56566591874379821537916830576, −1.42291480824776091817026402485, −1.11456933102686322758731587252, −0.60334433333222168708839991967, −0.087464364595941041859177282238,
0.087464364595941041859177282238, 0.60334433333222168708839991967, 1.11456933102686322758731587252, 1.42291480824776091817026402485, 1.56566591874379821537916830576, 1.61049140143573995827314163094, 2.48449953552213712499924520481, 2.52464494927919617870615438511, 2.82379802445324777454984750692, 3.08725971063358345708231687965, 3.29489312991416412464782763490, 3.74671946656088102014515150209, 3.86233774080046789716250612311, 4.08237623411166074745723648960, 4.52431060928792218274667896972, 4.57977351298623338823676199041, 4.76040216370376423020191417908, 4.87967084877586384608896834488, 5.20873921394203099062030772232, 5.32343922505680147297635481476, 5.52463584195135138291247305371, 5.71520306341466831760459369912, 6.03538927915580055116666514504, 6.30841562512505362466084337100, 6.32647727843282047794853010092
