| L(s) = 1 | + 8·13-s − 88·25-s + 56·37-s + 140·49-s − 40·61-s + 232·73-s − 248·97-s + 560·109-s + 248·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 596·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 8/13·13-s − 3.51·25-s + 1.51·37-s + 20/7·49-s − 0.655·61-s + 3.17·73-s − 2.55·97-s + 5.13·109-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 − 3.52·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.349930170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.349930170\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 7 | \( ( 1 - 10 p T^{2} + 5307 T^{4} - 10 p^{5} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 124 T^{2} + 15126 T^{4} - 124 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 2 T + 159 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 140 T^{2} + 136662 T^{4} + 140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 695 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 604 T^{2} + 632886 T^{4} - 604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2468 T^{2} + 2752998 T^{4} + 2468 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 1540 T^{2} + 1702662 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 14 T + 2607 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3140 T^{2} + 5167302 T^{4} + 3140 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 4516 T^{2} + 10784166 T^{4} - 4516 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4444 T^{2} + 14436726 T^{4} - 4444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 508 T^{2} + 14912358 T^{4} - 508 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 6076 T^{2} + 23942166 T^{4} - 6076 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 10 T + 7287 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8110 T^{2} + 40110387 T^{4} - 8110 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 292 T^{2} + 42446598 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 58 T + 10779 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 934 T^{2} - 28603749 T^{4} - 934 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 26116 T^{2} + 265140006 T^{4} - 26116 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 30668 T^{2} + 360580758 T^{4} + 30668 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 62 T + 8259 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−3.66802795776475342097279068223, −3.52663222329274984483614351819, −3.50045326656312215665885232592, −3.43245641244953114626264210812, −3.38014822770323754412145928936, −3.15280205744322308800391003962, −2.95542134380065904348034268392, −2.82272087929194240762919502506, −2.52124337167799671472798397534, −2.42522883912501252215764308263, −2.40115800492420652473744675353, −2.27297692732726088702685434292, −2.21552716187713104603465237312, −2.19154666866054014823793233369, −1.82250269592757573877161536328, −1.67091772239165505065201715217, −1.51023446928784991392968664405, −1.48231996053303544900358696692, −1.18730474776151180406231443087, −0.972779386954396924452800512981, −0.951801400777373354554397282505, −0.66791880290664058829360253537, −0.42275805772104576452421435694, −0.38417128212746178585976049721, −0.18691354623312625689894161684,
0.18691354623312625689894161684, 0.38417128212746178585976049721, 0.42275805772104576452421435694, 0.66791880290664058829360253537, 0.951801400777373354554397282505, 0.972779386954396924452800512981, 1.18730474776151180406231443087, 1.48231996053303544900358696692, 1.51023446928784991392968664405, 1.67091772239165505065201715217, 1.82250269592757573877161536328, 2.19154666866054014823793233369, 2.21552716187713104603465237312, 2.27297692732726088702685434292, 2.40115800492420652473744675353, 2.42522883912501252215764308263, 2.52124337167799671472798397534, 2.82272087929194240762919502506, 2.95542134380065904348034268392, 3.15280205744322308800391003962, 3.38014822770323754412145928936, 3.43245641244953114626264210812, 3.50045326656312215665885232592, 3.52663222329274984483614351819, 3.66802795776475342097279068223
Plot not available for L-functions of degree greater than 10.
