| L(s) = 1 | + 4·9-s + 40·25-s − 344·49-s + 592·73-s − 150·81-s − 496·97-s − 248·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 376·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 160·225-s + ⋯ |
| L(s) = 1 | + 4/9·9-s + 8/5·25-s − 7.02·49-s + 8.10·73-s − 1.85·81-s − 5.11·97-s − 2.04·121-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 − 2.22·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.711·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.708511343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.708511343\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 2 p T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 + 86 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 94 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 - p T )^{8}( 1 + p T )^{8} \) |
| 19 | \( ( 1 - 706 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 34 T + p^{2} T^{2} )^{4}( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 29 | \( ( 1 + 1622 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 + 1334 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1538 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 62 T + p^{2} T^{2} )^{4}( 1 + 62 T + p^{2} T^{2} )^{4} \) |
| 43 | \( ( 1 - 994 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 578 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 2678 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 5342 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 7394 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8194 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 9122 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 74 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 + 9782 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 - 12962 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 62 T + p^{2} T^{2} )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.48731854788779466688616558412, −5.13190203705690641423248937835, −5.08033902783299316075689974768, −4.90100494894744230906227999652, −4.83291195452269617949696274716, −4.78718711575287386014846420144, −4.69637679160806426485606922100, −4.36543668618635983248692288261, −3.96103935843980726437905307765, −3.83450297460821038863279822891, −3.75011821125977321651615241000, −3.72867524643368045531886266905, −3.53159305711022794165774671400, −3.21082067578087931213078360484, −2.84773487435238986734590973301, −2.71502667583170812685042756647, −2.69993230528348335204375209185, −2.64385560780302644313124233912, −2.01391619887246702822999560865, −1.71177872194694823770892995444, −1.66792950959333713365548033285, −1.46700698584765157619327395438, −1.10903924047594198540369993864, −0.51562194817120724260139913795, −0.38569749428616685395493355647,
0.38569749428616685395493355647, 0.51562194817120724260139913795, 1.10903924047594198540369993864, 1.46700698584765157619327395438, 1.66792950959333713365548033285, 1.71177872194694823770892995444, 2.01391619887246702822999560865, 2.64385560780302644313124233912, 2.69993230528348335204375209185, 2.71502667583170812685042756647, 2.84773487435238986734590973301, 3.21082067578087931213078360484, 3.53159305711022794165774671400, 3.72867524643368045531886266905, 3.75011821125977321651615241000, 3.83450297460821038863279822891, 3.96103935843980726437905307765, 4.36543668618635983248692288261, 4.69637679160806426485606922100, 4.78718711575287386014846420144, 4.83291195452269617949696274716, 4.90100494894744230906227999652, 5.08033902783299316075689974768, 5.13190203705690641423248937835, 5.48731854788779466688616558412
Plot not available for L-functions of degree greater than 10.
