| L(s) = 1 | + 6·7-s + 2·9-s − 4·17-s − 2·25-s − 8·31-s − 28·41-s − 8·47-s + 21·49-s + 12·63-s + 4·73-s − 24·79-s − 7·81-s + 4·89-s − 20·97-s + 8·103-s + 20·113-s − 24·119-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 2/3·9-s − 0.970·17-s − 2/5·25-s − 1.43·31-s − 4.37·41-s − 1.16·47-s + 3·49-s + 1.51·63-s + 0.468·73-s − 2.70·79-s − 7/9·81-s + 0.423·89-s − 2.03·97-s + 0.788·103-s + 1.88·113-s − 2.20·119-s + 2.72·121-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 − 0.646·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.859973778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.859973778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - T )^{6} \) |
| good | 3 | \( 1 - 2 T^{2} + 11 T^{4} - 20 T^{6} + 11 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 T^{2} + 59 T^{4} + 84 T^{6} + 59 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 30 T^{2} + 551 T^{4} - 7268 T^{6} + 551 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 + 2 T^{2} + 491 T^{4} + 660 T^{6} + 491 p^{2} T^{8} + 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 2 T + 23 T^{2} + 76 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 34 T^{2} + 907 T^{4} - 19540 T^{6} + 907 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 41 T^{2} + 16 T^{3} + 41 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( 1 - 82 T^{2} + 3719 T^{4} - 118812 T^{6} + 3719 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 31 | \( ( 1 + 4 T + 37 T^{2} - 8 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 98 T^{2} + 5239 T^{4} - 202172 T^{6} + 5239 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 14 T + 159 T^{2} + 1140 T^{3} + 159 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 62 T^{2} + 2023 T^{4} - 11300 T^{6} + 2023 p^{2} T^{8} - 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 4 T + 117 T^{2} + 312 T^{3} + 117 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 + 62 T^{2} + 10523 T^{4} + 399660 T^{6} + 10523 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 11659 T^{4} - 811724 T^{6} + 11659 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( 1 - 238 T^{2} + 28631 T^{4} - 2300292 T^{6} + 28631 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + p T^{2} )^{6} \) |
| 73 | \( ( 1 - 2 T + 151 T^{2} - 284 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( ( 1 + 12 T + 173 T^{2} + 1384 T^{3} + 173 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 274 T^{2} + 39467 T^{4} - 3784500 T^{6} + 39467 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 2 T + 151 T^{2} - 60 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 + 10 T + 103 T^{2} + 124 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−4.78009628614143026399284287745, −4.77153150291299208779165542666, −4.63980008495423788641195537033, −4.45564025527574278606092520639, −4.39474128722202801232121293555, −3.95652187145484919608445073356, −3.88744148353217125312799752818, −3.87116277185401175741840402565, −3.63843455905308069567328940906, −3.61563749805084134563405932839, −3.13540035918723197363166637823, −3.08055291753671013988237826554, −2.94342272301631178651416677331, −2.72620693710905562439493956044, −2.67038817358142031524296933063, −2.17781932329443526265331306934, −2.01579660570633269589244517561, −1.89532467675454826047043236856, −1.69394394341835347945538518818, −1.63892649321759467651755725620, −1.52087772071655196221635976317, −1.38845038030692251156954246773, −0.74564730519359422836459060412, −0.56496143115951535556432330435, −0.28280344569666110637029591097,
0.28280344569666110637029591097, 0.56496143115951535556432330435, 0.74564730519359422836459060412, 1.38845038030692251156954246773, 1.52087772071655196221635976317, 1.63892649321759467651755725620, 1.69394394341835347945538518818, 1.89532467675454826047043236856, 2.01579660570633269589244517561, 2.17781932329443526265331306934, 2.67038817358142031524296933063, 2.72620693710905562439493956044, 2.94342272301631178651416677331, 3.08055291753671013988237826554, 3.13540035918723197363166637823, 3.61563749805084134563405932839, 3.63843455905308069567328940906, 3.87116277185401175741840402565, 3.88744148353217125312799752818, 3.95652187145484919608445073356, 4.39474128722202801232121293555, 4.45564025527574278606092520639, 4.63980008495423788641195537033, 4.77153150291299208779165542666, 4.78009628614143026399284287745
Plot not available for L-functions of degree greater than 10.
