| 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
| \(L(1)\) |
\(\approx\) |
\(26.02245174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.02245174\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 6286 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−5.98744867663181494478613199548, −5.61195525274651024987528710319, −5.57840344345731255775798798352, −5.43738958023535851727039908601, −5.32909346551212528587484424485, −4.80579747490104092016538662075, −4.73157634080827542054283438549, −4.61495633903138081475506032159, −4.33802654249764818531529143622, −3.91785674939952230494127664973, −3.74776781978794530087447607966, −3.73759686176021479365313475180, −3.38479871937764520245286781257, −3.22324899175836172551946218979, −2.94567712307032330549647425733, −2.92203226714161181329521378041, −2.73836065319098562026578838415, −2.08776848751191137247460606318, −1.99649641577090893055883672527, −1.76981248079616675041469287305, −1.53045489176891164453057196729, −1.23150278463987248618962966705, −0.937060742615307666943291868182, −0.78869548348634475447640652247, −0.69952977551712132652291930951,
0.69952977551712132652291930951, 0.78869548348634475447640652247, 0.937060742615307666943291868182, 1.23150278463987248618962966705, 1.53045489176891164453057196729, 1.76981248079616675041469287305, 1.99649641577090893055883672527, 2.08776848751191137247460606318, 2.73836065319098562026578838415, 2.92203226714161181329521378041, 2.94567712307032330549647425733, 3.22324899175836172551946218979, 3.38479871937764520245286781257, 3.73759686176021479365313475180, 3.74776781978794530087447607966, 3.91785674939952230494127664973, 4.33802654249764818531529143622, 4.61495633903138081475506032159, 4.73157634080827542054283438549, 4.80579747490104092016538662075, 5.32909346551212528587484424485, 5.43738958023535851727039908601, 5.57840344345731255775798798352, 5.61195525274651024987528710319, 5.98744867663181494478613199548
