| L(s) = 1 | − 4-s − 5·9-s − 4·11-s + 16-s + 16·19-s + 12·29-s + 2·31-s + 5·36-s + 12·41-s + 4·44-s − 4·49-s − 18·59-s + 20·61-s − 5·64-s − 6·71-s − 16·76-s + 28·79-s + 9·81-s − 6·89-s + 20·99-s + 24·101-s + 40·109-s − 12·116-s + 10·121-s − 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 5/3·9-s − 1.20·11-s + 1/4·16-s + 3.67·19-s + 2.22·29-s + 0.359·31-s + 5/6·36-s + 1.87·41-s + 0.603·44-s − 4/7·49-s − 2.34·59-s + 2.56·61-s − 5/8·64-s − 0.712·71-s − 1.83·76-s + 3.15·79-s + 81-s − 0.635·89-s + 2.01·99-s + 2.38·101-s + 3.83·109-s − 1.11·116-s + 0.909·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.446954490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.446954490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $D_4$ | \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 846 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 85 T^{2} + 2856 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6006 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 9 T + 130 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 181 T^{2} + 15312 T^{4} + 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 337 T^{2} + 47136 T^{4} + 337 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.805944704445035951743390195337, −8.272881230858673627386982910731, −8.163877458247526013046625358900, −7.969254764108029927888327857103, −7.66958709093115142712351853110, −7.43447029750369811759003560764, −7.27109929736044219636591542524, −6.89306382559290384072921290535, −6.49791399336782193921578064496, −6.16179485315492020610358688499, −5.81942133877269352418057447655, −5.79040574221835326841680827982, −5.48454886309197426410481783222, −4.99509813568586081726323780789, −4.97169851862826787081390191426, −4.57594653956355275183647606167, −4.48242608704744081604113038338, −3.48814050069058789659644163215, −3.46283031192327173225695970565, −3.28398497542647616073534086748, −2.68263455343047363091290386036, −2.66292560046751736284570922934, −2.07253231868315618277971331391, −1.06914708103473513745467390536, −0.74265648109034133558651750304,
0.74265648109034133558651750304, 1.06914708103473513745467390536, 2.07253231868315618277971331391, 2.66292560046751736284570922934, 2.68263455343047363091290386036, 3.28398497542647616073534086748, 3.46283031192327173225695970565, 3.48814050069058789659644163215, 4.48242608704744081604113038338, 4.57594653956355275183647606167, 4.97169851862826787081390191426, 4.99509813568586081726323780789, 5.48454886309197426410481783222, 5.79040574221835326841680827982, 5.81942133877269352418057447655, 6.16179485315492020610358688499, 6.49791399336782193921578064496, 6.89306382559290384072921290535, 7.27109929736044219636591542524, 7.43447029750369811759003560764, 7.66958709093115142712351853110, 7.969254764108029927888327857103, 8.163877458247526013046625358900, 8.272881230858673627386982910731, 8.805944704445035951743390195337
