| L(s) = 1 | − 8·13-s + 16·25-s + 32·37-s + 8·49-s − 16·61-s + 32·73-s + 32·97-s − 8·109-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.21·13-s + 16/5·25-s + 5.26·37-s + 8/7·49-s − 2.04·61-s + 3.74·73-s + 3.24·97-s − 0.766·109-s − 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 5.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.792705490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.792705490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 56 T^{2} + 1335 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 32 T^{2} + 1263 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 8 T + 63 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 100 T^{2} + 5226 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 92 T^{2} + 5190 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T + 99 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 32 T^{2} - 5214 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 292 T^{2} + 33690 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 128 T^{2} + 19263 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.19080605051892410637337959407, −3.16454182488104602884074165298, −3.01853893218910670922393431984, −2.93826794862345761284574374517, −2.87991163518950760628658244394, −2.73958029911932714973278203875, −2.69320521160830037317076914765, −2.64651451109489083384823548517, −2.52932834048097775465352686581, −2.44216355021124849962569254451, −2.14046674446860863115067584714, −2.10412268618227782525053405349, −2.04109684156621253709743650539, −1.95548684326897528426836571371, −1.67841057747346964408671768241, −1.63277759033510142097543021403, −1.33993903947947778338480870364, −1.31729019161903690732244835045, −1.07579108696562078678586088236, −0.939064490158055530041713889477, −0.76403192186847222709481275126, −0.64696690545227095144025423804, −0.61524124936037310525260812396, −0.50863299088950066186046027218, −0.06568974814316189885431804681,
0.06568974814316189885431804681, 0.50863299088950066186046027218, 0.61524124936037310525260812396, 0.64696690545227095144025423804, 0.76403192186847222709481275126, 0.939064490158055530041713889477, 1.07579108696562078678586088236, 1.31729019161903690732244835045, 1.33993903947947778338480870364, 1.63277759033510142097543021403, 1.67841057747346964408671768241, 1.95548684326897528426836571371, 2.04109684156621253709743650539, 2.10412268618227782525053405349, 2.14046674446860863115067584714, 2.44216355021124849962569254451, 2.52932834048097775465352686581, 2.64651451109489083384823548517, 2.69320521160830037317076914765, 2.73958029911932714973278203875, 2.87991163518950760628658244394, 2.93826794862345761284574374517, 3.01853893218910670922393431984, 3.16454182488104602884074165298, 3.19080605051892410637337959407
Plot not available for L-functions of degree greater than 10.
