| L(s) = 1 | + 8·3-s + 16·9-s + 136·23-s + 20·25-s − 88·27-s + 64·31-s − 48·37-s − 152·47-s + 80·49-s + 352·53-s − 80·59-s − 24·67-s + 1.08e3·69-s + 256·71-s + 160·75-s − 644·81-s − 368·89-s + 512·93-s − 32·97-s + 56·103-s − 384·111-s − 416·113-s + 132·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 8/3·3-s + 16/9·9-s + 5.91·23-s + 4/5·25-s − 3.25·27-s + 2.06·31-s − 1.29·37-s − 3.23·47-s + 1.63·49-s + 6.64·53-s − 1.35·59-s − 0.358·67-s + 15.7·69-s + 3.60·71-s + 2.13·75-s − 7.95·81-s − 4.13·89-s + 5.50·93-s − 0.329·97-s + 0.543·103-s − 3.45·111-s − 3.68·113-s + 1.09·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(32.81617267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(32.81617267\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 11 | \( 1 - 12 p T^{2} + 18 p^{3} T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} \) |
| good | 3 | \( ( 1 - 4 T + 16 T^{2} - 20 T^{3} + 50 T^{4} - 20 p^{2} T^{5} + 16 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 80 T^{2} + 932 p T^{4} - 325040 T^{6} + 19002246 T^{8} - 325040 p^{4} T^{10} + 932 p^{9} T^{12} - 80 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 - 456 T^{2} + 114260 T^{4} - 22854744 T^{6} + 3991733734 T^{8} - 22854744 p^{4} T^{10} + 114260 p^{8} T^{12} - 456 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( 1 - 1256 T^{2} + 691476 T^{4} - 234999352 T^{6} + 67056219686 T^{8} - 234999352 p^{4} T^{10} + 691476 p^{8} T^{12} - 1256 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( 1 - 1224 T^{2} + 751420 T^{4} - 385041016 T^{6} + 163401263174 T^{8} - 385041016 p^{4} T^{10} + 751420 p^{8} T^{12} - 1224 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( ( 1 - 68 T + 2224 T^{2} - 36724 T^{3} + 584146 T^{4} - 36724 p^{2} T^{5} + 2224 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 - 4616 T^{2} + 10591964 T^{4} - 15445127096 T^{6} + 15489128469510 T^{8} - 15445127096 p^{4} T^{10} + 10591964 p^{8} T^{12} - 4616 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 32 T + 2652 T^{2} - 75808 T^{3} + 3628278 T^{4} - 75808 p^{2} T^{5} + 2652 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 24 T + 5276 T^{2} + 91880 T^{3} + 10665510 T^{4} + 91880 p^{2} T^{5} + 5276 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 2712 T^{2} + 10630844 T^{4} - 17663882920 T^{6} + 41573097239430 T^{8} - 17663882920 p^{4} T^{10} + 10630844 p^{8} T^{12} - 2712 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( 1 - 7984 T^{2} + 32118524 T^{4} - 88434596304 T^{6} + 185487043452870 T^{8} - 88434596304 p^{4} T^{10} + 32118524 p^{8} T^{12} - 7984 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( ( 1 + 76 T + 5216 T^{2} + 277340 T^{3} + 13396690 T^{4} + 277340 p^{2} T^{5} + 5216 p^{4} T^{6} + 76 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 176 T + 21212 T^{2} - 1660368 T^{3} + 103199910 T^{4} - 1660368 p^{2} T^{5} + 21212 p^{4} T^{6} - 176 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( ( 1 + 40 T + 9748 T^{2} + 498680 T^{3} + 731762 p T^{4} + 498680 p^{2} T^{5} + 9748 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 6040 T^{2} + 27425148 T^{4} - 79174240040 T^{6} + 356791480770758 T^{8} - 79174240040 p^{4} T^{10} + 27425148 p^{8} T^{12} - 6040 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 + 12 T + 9920 T^{2} + 218172 T^{3} + 49331314 T^{4} + 218172 p^{2} T^{5} + 9920 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 128 T + 18052 T^{2} - 1583232 T^{3} + 138951878 T^{4} - 1583232 p^{2} T^{5} + 18052 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 73 | \( 1 - 4488 T^{2} + 81810708 T^{4} - 275881335576 T^{6} + 3118297538431910 T^{8} - 275881335576 p^{4} T^{10} + 81810708 p^{8} T^{12} - 4488 p^{12} T^{14} + p^{16} T^{16} \) |
| 79 | \( 1 - 23944 T^{2} + 186312316 T^{4} - 99649203768 T^{6} - 4663183270460794 T^{8} - 99649203768 p^{4} T^{10} + 186312316 p^{8} T^{12} - 23944 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 16144 T^{2} + 111995964 T^{4} - 818543769584 T^{6} + 6744844911959750 T^{8} - 818543769584 p^{4} T^{10} + 111995964 p^{8} T^{12} - 16144 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 184 T + 21524 T^{2} + 1384744 T^{3} + 114465510 T^{4} + 1384744 p^{2} T^{5} + 21524 p^{4} T^{6} + 184 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 124 p T^{2} + 335856 T^{3} + 188698118 T^{4} + 335856 p^{2} T^{5} + 124 p^{5} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−4.11192894309639850611753212835, −3.93772724642397647516762134632, −3.78227723331065384497756434373, −3.76107781192582825916268043556, −3.33356032051949083235110749430, −3.33092609872391443403002885028, −3.24750483782588678385637641335, −3.15329005619805196684264238972, −3.05756487763152610845127580836, −2.98808871744196009211143816500, −2.63399716138650458001327506473, −2.55208308179361631086020213608, −2.52200436872098428549909446430, −2.51330318143138063208550783096, −2.42290490982442113422695322014, −1.91417148535558647514893524750, −1.81559215652423149754039217583, −1.73892263072373014253402437675, −1.54507062998865902430020733415, −1.12494412064164457603229611924, −1.03018583267205470287562308637, −0.862436417303307714273948299115, −0.831278614453935666422324266787, −0.45047255545888444479670962289, −0.25314019173358335593758934509,
0.25314019173358335593758934509, 0.45047255545888444479670962289, 0.831278614453935666422324266787, 0.862436417303307714273948299115, 1.03018583267205470287562308637, 1.12494412064164457603229611924, 1.54507062998865902430020733415, 1.73892263072373014253402437675, 1.81559215652423149754039217583, 1.91417148535558647514893524750, 2.42290490982442113422695322014, 2.51330318143138063208550783096, 2.52200436872098428549909446430, 2.55208308179361631086020213608, 2.63399716138650458001327506473, 2.98808871744196009211143816500, 3.05756487763152610845127580836, 3.15329005619805196684264238972, 3.24750483782588678385637641335, 3.33092609872391443403002885028, 3.33356032051949083235110749430, 3.76107781192582825916268043556, 3.78227723331065384497756434373, 3.93772724642397647516762134632, 4.11192894309639850611753212835
Plot not available for L-functions of degree greater than 10.
