L(s) = 1 | − 4·9-s + 16·17-s − 48·41-s − 32·73-s + 10·81-s − 80·89-s + 48·97-s + 48·113-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·153-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | − 4/3·9-s + 3.88·17-s − 7.49·41-s − 3.74·73-s + 10/9·81-s − 8.47·89-s + 4.87·97-s + 4.51·113-s + 3.63·121-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 − 5.17·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.426217824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.426217824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{4} \) |
good | 5 | \( ( 1 + 18 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 30 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 12 T^{2} + 246 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 12 T^{2} + 582 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 1650 T^{4} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 96 T^{2} + 4098 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 92 T^{2} + 4342 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 4 T^{2} - 906 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 108 T^{2} + 6822 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 64 T^{2} + 1234 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 140 T^{2} + 9814 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 188 T^{2} + 15766 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 76 T^{2} + 2230 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 172 T^{2} + 15430 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 + 288 T^{2} + 33090 T^{4} + 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 244 T^{2} + 27510 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 10 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.86868175713437682495495675214, −3.75799880549871706845291518964, −3.68137890389513033364219019005, −3.40145011658186797323700452659, −3.37202306975781035721045574643, −3.32943793879108114115793763660, −3.30705778276571865386709057276, −3.17920525369256454144286557693, −3.12414640367897992340790257158, −3.05686248787806545164710262175, −2.69226445943583969389480974557, −2.57585877386610981717082450740, −2.49479590549808739686846296922, −2.23808516812216536091062178523, −2.18694414754632655103660959913, −1.82562452023996376685417625620, −1.68981595512792203993125454470, −1.55720205192910610507330168718, −1.55351942445669601712156309074, −1.48317428202289539014438016199, −1.13066910219665962873363184872, −0.954658607840750269847251182886, −0.62892016868338844535643321797, −0.39766115295678557302511248842, −0.15069980009300507145075836019,
0.15069980009300507145075836019, 0.39766115295678557302511248842, 0.62892016868338844535643321797, 0.954658607840750269847251182886, 1.13066910219665962873363184872, 1.48317428202289539014438016199, 1.55351942445669601712156309074, 1.55720205192910610507330168718, 1.68981595512792203993125454470, 1.82562452023996376685417625620, 2.18694414754632655103660959913, 2.23808516812216536091062178523, 2.49479590549808739686846296922, 2.57585877386610981717082450740, 2.69226445943583969389480974557, 3.05686248787806545164710262175, 3.12414640367897992340790257158, 3.17920525369256454144286557693, 3.30705778276571865386709057276, 3.32943793879108114115793763660, 3.37202306975781035721045574643, 3.40145011658186797323700452659, 3.68137890389513033364219019005, 3.75799880549871706845291518964, 3.86868175713437682495495675214
Plot not available for L-functions of degree greater than 10.