| L(s) = 1 | − 4·5-s − 12·13-s + 16·17-s + 8·25-s − 4·29-s − 12·37-s + 12·49-s + 20·53-s + 4·61-s + 48·65-s − 64·85-s + 64·97-s − 44·101-s + 12·109-s − 72·113-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 3.32·13-s + 3.88·17-s + 8/5·25-s − 0.742·29-s − 1.97·37-s + 12/7·49-s + 2.74·53-s + 0.512·61-s + 5.95·65-s − 6.94·85-s + 6.49·97-s − 4.37·101-s + 1.14·109-s − 6.77·113-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5310704438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5310704438\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 46 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 6286 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
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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
−5.77867804553742477658966156945, −5.57096322261759146912903148444, −5.45837983223693093134347790748, −5.10228857731272582292915049574, −5.06095697506380437114015021701, −5.03586835131962941871618038098, −4.90324388438020718973604069557, −4.38441962293940955619382045412, −4.33611468500528676516464733921, −3.88967481323604341514893060386, −3.85261575663729797541386393647, −3.63665450064260422775897779337, −3.59649741340047339611696488138, −3.34835596246754841484327181432, −3.06756549902876953929241453046, −2.68740095010459243502948933940, −2.47995761720607518086329589089, −2.45016777890146802801002543765, −2.33360363840436450258216981460, −1.76779212599054284470929543457, −1.38624254450645157966379726734, −1.13478804443325272994465645086, −1.01398210350718607710114816528, −0.44877821206095494015160075251, −0.15796444523906369776100522032,
0.15796444523906369776100522032, 0.44877821206095494015160075251, 1.01398210350718607710114816528, 1.13478804443325272994465645086, 1.38624254450645157966379726734, 1.76779212599054284470929543457, 2.33360363840436450258216981460, 2.45016777890146802801002543765, 2.47995761720607518086329589089, 2.68740095010459243502948933940, 3.06756549902876953929241453046, 3.34835596246754841484327181432, 3.59649741340047339611696488138, 3.63665450064260422775897779337, 3.85261575663729797541386393647, 3.88967481323604341514893060386, 4.33611468500528676516464733921, 4.38441962293940955619382045412, 4.90324388438020718973604069557, 5.03586835131962941871618038098, 5.06095697506380437114015021701, 5.10228857731272582292915049574, 5.45837983223693093134347790748, 5.57096322261759146912903148444, 5.77867804553742477658966156945
