| L(s) = 1 | − 4·4-s + 4·13-s + 12·16-s + 16·19-s − 28·31-s − 8·37-s + 28·43-s + 10·49-s − 16·52-s + 40·61-s − 32·64-s + 4·67-s − 64·76-s + 8·79-s − 28·97-s + 40·109-s + 112·124-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.10·13-s + 3·16-s + 3.67·19-s − 5.02·31-s − 1.31·37-s + 4.26·43-s + 10/7·49-s − 2.21·52-s + 5.12·61-s − 4·64-s + 0.488·67-s − 7.34·76-s + 0.900·79-s − 2.84·97-s + 3.83·109-s + 10.0·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613359930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613359930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 31 T^{4} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 233 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )( 1 + 40 T^{2} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 5407 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 6638 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 11753 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 176 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.189647505455766750007102105390, −7.77621427101332911067581243297, −7.65763996633432484377989348785, −7.36904087970834219375644990118, −7.15257645919197829701276295688, −6.99280715791927028667814909058, −6.81752404760477237939586679878, −6.10129139749155499702450030233, −5.79636093441933437442307283107, −5.56852503776266656096625137062, −5.50197049362225323294404543938, −5.34424065956781770844509271726, −5.31126208466626089564556919339, −4.70251855247389179231846771630, −4.42636901150732962099004719497, −3.90117756379454380132469208167, −3.76483011059339621029102631114, −3.65042521964223451480747970573, −3.53168370886658310772605114420, −3.04983304176838091241052994574, −2.48700925966976332164969174878, −2.10015245681796227737590973184, −1.42832791712060923972908294814, −0.999190478304177383108947652588, −0.64392064367176100058651411442,
0.64392064367176100058651411442, 0.999190478304177383108947652588, 1.42832791712060923972908294814, 2.10015245681796227737590973184, 2.48700925966976332164969174878, 3.04983304176838091241052994574, 3.53168370886658310772605114420, 3.65042521964223451480747970573, 3.76483011059339621029102631114, 3.90117756379454380132469208167, 4.42636901150732962099004719497, 4.70251855247389179231846771630, 5.31126208466626089564556919339, 5.34424065956781770844509271726, 5.50197049362225323294404543938, 5.56852503776266656096625137062, 5.79636093441933437442307283107, 6.10129139749155499702450030233, 6.81752404760477237939586679878, 6.99280715791927028667814909058, 7.15257645919197829701276295688, 7.36904087970834219375644990118, 7.65763996633432484377989348785, 7.77621427101332911067581243297, 8.189647505455766750007102105390
