L(s) = 1 | + 2·5-s + 4·13-s − 4·17-s + 3·25-s + 4·29-s + 4·37-s + 4·41-s − 2·49-s − 20·53-s + 20·61-s + 8·65-s + 4·73-s − 8·85-s + 12·89-s + 4·97-s + 4·101-s + 20·109-s − 4·113-s + 10·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.10·13-s − 0.970·17-s + 3/5·25-s + 0.742·29-s + 0.657·37-s + 0.624·41-s − 2/7·49-s − 2.74·53-s + 2.56·61-s + 0.992·65-s + 0.468·73-s − 0.867·85-s + 1.27·89-s + 0.406·97-s + 0.398·101-s + 1.91·109-s − 0.376·113-s + 0.909·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.397583136\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.397583136\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.582434093328023039487427520843, −8.175059731738367095390236775908, −7.57779509096492161138851961752, −7.09761933289016537045263321881, −6.40514922838054858362750919600, −6.29206512724321849355908480041, −5.92132532306432686373161032023, −5.09965270463121083426444288385, −4.85967652801373793418170645446, −4.18120542037032132882087257829, −3.61565482393410745408233101998, −2.97702695890457152557394603586, −2.31711616556542466777401422408, −1.72256120653298784132741933027, −0.844561202731234930629342791772,
0.844561202731234930629342791772, 1.72256120653298784132741933027, 2.31711616556542466777401422408, 2.97702695890457152557394603586, 3.61565482393410745408233101998, 4.18120542037032132882087257829, 4.85967652801373793418170645446, 5.09965270463121083426444288385, 5.92132532306432686373161032023, 6.29206512724321849355908480041, 6.40514922838054858362750919600, 7.09761933289016537045263321881, 7.57779509096492161138851961752, 8.175059731738367095390236775908, 8.582434093328023039487427520843