L(s) = 1 | + 4·13-s + 4·17-s − 12·29-s + 20·37-s − 4·41-s − 6·49-s + 12·53-s − 4·61-s + 12·73-s − 20·89-s − 4·97-s + 4·101-s − 36·109-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 1.10·13-s + 0.970·17-s − 2.22·29-s + 3.28·37-s − 0.624·41-s − 6/7·49-s + 1.64·53-s − 0.512·61-s + 1.40·73-s − 2.11·89-s − 0.406·97-s + 0.398·101-s − 3.44·109-s + 0.376·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.067066103\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.067066103\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.963679046920402532264407763238, −7.888741344818032251001908181943, −7.29032545485371000584885007411, −7.28184996764748464837294750971, −6.51712836887501865120675784078, −6.46493350809301987933909227990, −5.91460594466987383152551432696, −5.67951365770390818258446695246, −5.37717874735943276420743381115, −5.04445240460816739377206551611, −4.26654561517935832143389712215, −4.24128894933764796336293531791, −3.74614386642154963353639196215, −3.46676984338734785644367617634, −2.80438642072844496316530101248, −2.69063590046686880840196279489, −1.91656597738313881477065527347, −1.56877042670706713648182220266, −1.01382778911586294831395181025, −0.48183596480629773473459067339,
0.48183596480629773473459067339, 1.01382778911586294831395181025, 1.56877042670706713648182220266, 1.91656597738313881477065527347, 2.69063590046686880840196279489, 2.80438642072844496316530101248, 3.46676984338734785644367617634, 3.74614386642154963353639196215, 4.24128894933764796336293531791, 4.26654561517935832143389712215, 5.04445240460816739377206551611, 5.37717874735943276420743381115, 5.67951365770390818258446695246, 5.91460594466987383152551432696, 6.46493350809301987933909227990, 6.51712836887501865120675784078, 7.28184996764748464837294750971, 7.29032545485371000584885007411, 7.888741344818032251001908181943, 7.963679046920402532264407763238