| L(s) = 1 | − 4·7-s − 5·13-s + 19-s + 5·25-s + 22·31-s − 11·37-s + 13·43-s + 9·49-s + 28·61-s + 10·67-s − 17·73-s + 34·79-s + 20·91-s − 14·97-s + 13·103-s + 19·109-s + 11·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.38·13-s + 0.229·19-s + 25-s + 3.95·31-s − 1.80·37-s + 1.98·43-s + 9/7·49-s + 3.58·61-s + 1.22·67-s − 1.98·73-s + 3.82·79-s + 2.09·91-s − 1.42·97-s + 1.28·103-s + 1.81·109-s + 121-s + 0.0887·127-s + 0.0873·131-s − 0.346·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.942424221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942424221\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.181868544056807885739837005070, −8.930898047012488564777943986352, −8.326574508146013813960542586572, −8.269988843952763425647484544570, −7.48306579237977361422448881245, −7.35129342344752546962902020720, −6.68045429032085766302893408214, −6.60433534089958663213161479259, −6.32615023441854020256931723144, −5.62273514428214810333857452246, −5.25729544955028738795297966033, −4.86917905983158588302190402458, −4.37725736147734483757950110530, −3.96593442636860419356498160942, −3.27972309189718058223262044643, −3.04519014082006604841771215464, −2.37307012056556679713299820724, −2.27926010680838742970398790761, −0.938753732821661070228653948191, −0.61311950900277576780674795783,
0.61311950900277576780674795783, 0.938753732821661070228653948191, 2.27926010680838742970398790761, 2.37307012056556679713299820724, 3.04519014082006604841771215464, 3.27972309189718058223262044643, 3.96593442636860419356498160942, 4.37725736147734483757950110530, 4.86917905983158588302190402458, 5.25729544955028738795297966033, 5.62273514428214810333857452246, 6.32615023441854020256931723144, 6.60433534089958663213161479259, 6.68045429032085766302893408214, 7.35129342344752546962902020720, 7.48306579237977361422448881245, 8.269988843952763425647484544570, 8.326574508146013813960542586572, 8.930898047012488564777943986352, 9.181868544056807885739837005070
