| L(s) = 1 | − 5·4-s − 3·16-s + 80·29-s − 136·41-s − 120·49-s − 224·61-s + 75·64-s − 440·89-s + 544·101-s − 800·109-s − 400·116-s + 708·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 680·164-s + 167-s + 680·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 5/4·4-s − 0.187·16-s + 2.75·29-s − 3.31·41-s − 2.44·49-s − 3.67·61-s + 1.17·64-s − 4.94·89-s + 5.38·101-s − 7.33·109-s − 3.44·116-s + 5.85·121-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 + 4.14·164-s + 0.00598·167-s + 4.02·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.232493099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.232493099\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 5 T^{2} + 7 p^{2} T^{4} + 5 p^{4} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 60 T^{2} + 3078 T^{4} + 60 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 - 354 T^{2} + 56511 T^{4} - 354 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 340 T^{2} + 83398 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 810 T^{2} + 325163 T^{4} - 810 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 74 T^{2} + 257911 T^{4} - 74 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 1340 T^{2} + 1005958 T^{4} + 1340 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 - 20 T + 1618 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 3324 T^{2} + 4543686 T^{4} - 3324 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 3100 T^{2} + 5479078 T^{4} - 3100 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 17 T + p^{2} T^{2} )^{8} \) |
| 43 | \( ( 1 + 4340 T^{2} + 9339718 T^{4} + 4340 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 3860 T^{2} + 12536998 T^{4} + 3860 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3100 T^{2} + 3027238 T^{4} - 3100 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 - 244 T^{2} + 18936006 T^{4} - 244 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 + 56 T + 4126 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 + 14370 T^{2} + 90627423 T^{4} + 14370 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 1044 T^{2} + 27414246 T^{4} - 1044 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 14490 T^{2} + 107155163 T^{4} - 14490 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 17244 T^{2} + 144301446 T^{4} - 17244 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 + 22130 T^{2} + 213611503 T^{4} + 22130 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 110 T + 18703 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 13660 T^{2} + 80027718 T^{4} - 13660 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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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
−4.16070521290905087866267979211, −4.10064055466925188350435556980, −3.89677021281771341282494515375, −3.86286457030292526642718275895, −3.52904525244434184801037809797, −3.33599258847794613250205515497, −3.32833250701421017528218749306, −3.17293705579116063574941251925, −2.92981685907557912167059970714, −2.87217003215777695404485589388, −2.82703933087186834683200309175, −2.71975015559850094154722482743, −2.67787396352231454237284867120, −2.17389209982257091212655381492, −1.92448115846296829966435132199, −1.78420767407725856057893917574, −1.74482837855546867779378725189, −1.72744164792531293253973251330, −1.40067204441139573851562845954, −1.35693778327280884110582508218, −0.969598980089668065611412714304, −0.60610505303260894859361680659, −0.51873118085316633223227944769, −0.35344087511304212339605688959, −0.28055180028908727482558445510,
0.28055180028908727482558445510, 0.35344087511304212339605688959, 0.51873118085316633223227944769, 0.60610505303260894859361680659, 0.969598980089668065611412714304, 1.35693778327280884110582508218, 1.40067204441139573851562845954, 1.72744164792531293253973251330, 1.74482837855546867779378725189, 1.78420767407725856057893917574, 1.92448115846296829966435132199, 2.17389209982257091212655381492, 2.67787396352231454237284867120, 2.71975015559850094154722482743, 2.82703933087186834683200309175, 2.87217003215777695404485589388, 2.92981685907557912167059970714, 3.17293705579116063574941251925, 3.32833250701421017528218749306, 3.33599258847794613250205515497, 3.52904525244434184801037809797, 3.86286457030292526642718275895, 3.89677021281771341282494515375, 4.10064055466925188350435556980, 4.16070521290905087866267979211
Plot not available for L-functions of degree greater than 10.
