| L(s) = 1 | − 4·5-s − 3·9-s − 8·11-s + 5·25-s − 18·29-s + 4·31-s + 16·41-s + 12·45-s + 49-s + 32·55-s + 4·59-s + 20·61-s + 60·71-s − 40·89-s + 24·99-s − 28·101-s + 8·109-s + 38·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 72·145-s + 149-s + 151-s − 16·155-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 9-s − 2.41·11-s + 25-s − 3.34·29-s + 0.718·31-s + 2.49·41-s + 1.78·45-s + 1/7·49-s + 4.31·55-s + 0.520·59-s + 2.56·61-s + 7.12·71-s − 4.23·89-s + 2.41·99-s − 2.78·101-s + 0.766·109-s + 3.45·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.97·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7581689054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7581689054\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 17 T^{2} + 120 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 141 T^{2} + 12992 T^{4} + 141 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 + 94 T^{2} - 573 T^{4} + 94 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.98723521080913504598625588893, −6.86897342765211006097996292783, −6.51808465457385390205990271300, −6.15153611542406303321877028999, −6.12679294336890097457756204427, −5.53703103319753871263825235567, −5.44698358171523509653490043032, −5.43078662652655704820051166546, −5.37291433655454752979645746624, −4.86776518831482652007632499713, −4.82464740376244242357242108563, −4.19442792498723532775955895236, −4.13759729138203064746141066635, −3.83295036199417631668098564157, −3.76084035316019385047430603415, −3.63029367988947763109810121913, −2.97679726476481985213690496002, −2.92829581487723446817933879150, −2.63792625550613652716420046502, −2.24260433948137023612694767597, −2.22653841121348603991933940973, −1.77658687271086210975139644633, −1.03874758617165034622041014301, −0.52371962766200080314849693521, −0.34132588534484034179728540804,
0.34132588534484034179728540804, 0.52371962766200080314849693521, 1.03874758617165034622041014301, 1.77658687271086210975139644633, 2.22653841121348603991933940973, 2.24260433948137023612694767597, 2.63792625550613652716420046502, 2.92829581487723446817933879150, 2.97679726476481985213690496002, 3.63029367988947763109810121913, 3.76084035316019385047430603415, 3.83295036199417631668098564157, 4.13759729138203064746141066635, 4.19442792498723532775955895236, 4.82464740376244242357242108563, 4.86776518831482652007632499713, 5.37291433655454752979645746624, 5.43078662652655704820051166546, 5.44698358171523509653490043032, 5.53703103319753871263825235567, 6.12679294336890097457756204427, 6.15153611542406303321877028999, 6.51808465457385390205990271300, 6.86897342765211006097996292783, 6.98723521080913504598625588893
