| L(s) = 1 | + 4-s + 2·5-s − 3·9-s − 2·16-s − 4·19-s + 2·20-s + 25-s − 16·29-s + 16·31-s − 3·36-s + 16·41-s − 6·45-s + 10·49-s + 16·59-s − 4·61-s − 10·64-s − 24·71-s − 4·76-s − 12·79-s − 4·80-s + 6·81-s − 4·89-s − 8·95-s + 100-s + 80·101-s − 12·109-s − 16·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s − 9-s − 1/2·16-s − 0.917·19-s + 0.447·20-s + 1/5·25-s − 2.97·29-s + 2.87·31-s − 1/2·36-s + 2.49·41-s − 0.894·45-s + 10/7·49-s + 2.08·59-s − 0.512·61-s − 5/4·64-s − 2.84·71-s − 0.458·76-s − 1.35·79-s − 0.447·80-s + 2/3·81-s − 0.423·89-s − 0.820·95-s + 1/10·100-s + 7.96·101-s − 1.14·109-s − 1.48·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.742392115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.742392115\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T^{2} + 3 T^{4} + 5 T^{6} + 3 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 10 T^{2} + 31 T^{4} - 124 T^{6} + 31 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 14 T^{2} + 311 T^{4} + 4692 T^{6} + 311 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 30 T^{2} + 1151 T^{4} - 18164 T^{6} + 1151 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{3} \) |
| 29 | \( ( 1 + 8 T + 3 p T^{2} + 432 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 174 T^{2} + 13943 T^{4} - 655652 T^{6} + 13943 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 8 T + 3 p T^{2} - 624 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 226 T^{2} + 22423 T^{4} - 1251628 T^{6} + 22423 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 186 T^{2} + 17903 T^{4} - 1043180 T^{6} + 17903 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 190 T^{2} + 20375 T^{4} - 1316100 T^{6} + 20375 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 8 T + 113 T^{2} - 1024 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 2 T + 131 T^{2} + 204 T^{3} + 131 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 226 T^{2} + 28103 T^{4} - 2235900 T^{6} + 28103 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 12 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 234 T^{2} + 27279 T^{4} - 2258332 T^{6} + 27279 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 486 T^{2} + 99383 T^{4} - 10945028 T^{6} + 99383 p^{2} T^{8} - 486 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 255 T^{2} + 348 T^{3} + 255 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 454 T^{2} + 94543 T^{4} - 11606932 T^{6} + 94543 p^{2} T^{8} - 454 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.75875465007929563546123876387, −4.73319813848862756907583063950, −4.56785493210523371786209950610, −4.41056493132618463362505080502, −4.20789378806500242954852338710, −4.13455300207923968585552750565, −4.03316793998777281659075572231, −3.63480428267262917058677541689, −3.45799172365067693690704123452, −3.42233515490049185456623835207, −3.35917160604929187125475277808, −2.99508047073559081987587934972, −2.89545046993389894552480190875, −2.56011796082708756456504158936, −2.48181600999973805300676977255, −2.48037566203424642010793807237, −2.14226703114687434567043100894, −2.04715633323752515776129872103, −1.89425654314241031346827066404, −1.73743242145751510877392589600, −1.25587853126072217047620392194, −1.01781725162231601011668709410, −1.01169863256713078039844389528, −0.52456550461571399367251903672, −0.20967433896678796843368782995,
0.20967433896678796843368782995, 0.52456550461571399367251903672, 1.01169863256713078039844389528, 1.01781725162231601011668709410, 1.25587853126072217047620392194, 1.73743242145751510877392589600, 1.89425654314241031346827066404, 2.04715633323752515776129872103, 2.14226703114687434567043100894, 2.48037566203424642010793807237, 2.48181600999973805300676977255, 2.56011796082708756456504158936, 2.89545046993389894552480190875, 2.99508047073559081987587934972, 3.35917160604929187125475277808, 3.42233515490049185456623835207, 3.45799172365067693690704123452, 3.63480428267262917058677541689, 4.03316793998777281659075572231, 4.13455300207923968585552750565, 4.20789378806500242954852338710, 4.41056493132618463362505080502, 4.56785493210523371786209950610, 4.73319813848862756907583063950, 4.75875465007929563546123876387
Plot not available for L-functions of degree greater than 10.
