| L(s) = 1 | + 4·9-s + 20·25-s − 12·49-s + 96·79-s − 6·81-s + 16·109-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 80·225-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 4·25-s − 1.71·49-s + 10.8·79-s − 2/3·81-s + 1.53·109-s + 5.09·121-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 − 5.53·169-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 + 0.0669·223-s + 16/3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03699148380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03699148380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 11 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 13 | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.97695647846543462540607169458, −3.77456190702803313898999264450, −3.68501572266880822936778865183, −3.43857987365111204259667648568, −3.43487935312739711970479603958, −3.36247733962440794852700077235, −3.26340141177743512546061008896, −3.13724623489535226091506707634, −2.99537057406156677312071158608, −2.87389911155737722863972645981, −2.61932196197713247195347312369, −2.30550781879754993080931590867, −2.28892411867236106685395493748, −2.28859399386205283719148848316, −2.17562281461351829687214764199, −1.99074337841513132935628741962, −1.93938948868570216715942765963, −1.42366728225737860853336708296, −1.39236323519069076094302963271, −1.29885741253700159946830402837, −1.05178029210315359216148293467, −0.856516053484519909405234030264, −0.77135767691153766315896725513, −0.68146434087381136431041052809, −0.01348944501963307379158627196,
0.01348944501963307379158627196, 0.68146434087381136431041052809, 0.77135767691153766315896725513, 0.856516053484519909405234030264, 1.05178029210315359216148293467, 1.29885741253700159946830402837, 1.39236323519069076094302963271, 1.42366728225737860853336708296, 1.93938948868570216715942765963, 1.99074337841513132935628741962, 2.17562281461351829687214764199, 2.28859399386205283719148848316, 2.28892411867236106685395493748, 2.30550781879754993080931590867, 2.61932196197713247195347312369, 2.87389911155737722863972645981, 2.99537057406156677312071158608, 3.13724623489535226091506707634, 3.26340141177743512546061008896, 3.36247733962440794852700077235, 3.43487935312739711970479603958, 3.43857987365111204259667648568, 3.68501572266880822936778865183, 3.77456190702803313898999264450, 3.97695647846543462540607169458
Plot not available for L-functions of degree greater than 10.
