| L(s) = 1 | + 5·9-s − 6·19-s − 34·29-s + 4·31-s + 12·41-s + 17·49-s − 30·59-s + 16·61-s − 8·71-s − 4·79-s + 2·81-s − 12·89-s − 24·101-s − 38·109-s − 66·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 57·169-s − 30·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 1.37·19-s − 6.31·29-s + 0.718·31-s + 1.87·41-s + 17/7·49-s − 3.90·59-s + 2.04·61-s − 0.949·71-s − 0.450·79-s + 2/9·81-s − 1.27·89-s − 2.38·101-s − 3.63·109-s − 6·121-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 + 4.38·169-s − 2.29·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.661780061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.661780061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 + T )^{6} \) |
| good | 3 | \( 1 - 5 T^{2} + 23 T^{4} - 86 T^{6} + 23 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 17 T^{2} + 211 T^{4} - 1718 T^{6} + 211 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + p T^{2} )^{6} \) |
| 13 | \( 1 - 57 T^{2} + 1487 T^{4} - 23774 T^{6} + 1487 p^{2} T^{8} - 57 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 69 T^{2} + 2147 T^{4} - 42926 T^{6} + 2147 p^{2} T^{8} - 69 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 73 T^{2} + 3107 T^{4} - 85926 T^{6} + 3107 p^{2} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 17 T + 171 T^{2} + 1110 T^{3} + 171 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 2 T + 45 T^{2} + 4 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 11191 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 174 T^{2} + 13991 T^{4} - 717764 T^{6} + 13991 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 106 T^{2} + 5423 T^{4} - 232716 T^{6} + 5423 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 209 T^{2} + 20271 T^{4} - 1269614 T^{6} + 20271 p^{2} T^{8} - 209 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 15 T + 149 T^{2} + 986 T^{3} + 149 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 8 T + 179 T^{2} - 912 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 37 T^{2} + 7511 T^{4} - 169110 T^{6} + 7511 p^{2} T^{8} - 37 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 4 T + 21 T^{2} - 456 T^{3} + 21 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 229 T^{2} + 27155 T^{4} - 30366 p T^{6} + 27155 p^{2} T^{8} - 229 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 2 T + 213 T^{2} + 284 T^{3} + 213 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 366 T^{2} + 60407 T^{4} - 6127364 T^{6} + 60407 p^{2} T^{8} - 366 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 6 T + 215 T^{2} + 884 T^{3} + 215 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 358 T^{2} + 62671 T^{4} - 7159060 T^{6} + 62671 p^{2} T^{8} - 358 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.28796108798481282090230690788, −4.23071940256840709222236083897, −4.18499771242696888959210147481, −4.05960019480092712013542013190, −3.79803893725793442786446647294, −3.69165310339962767369560678964, −3.57495112561867474205829030122, −3.56428376748819566317412804000, −3.16697465462797393106678710151, −3.15188261646090351558425730707, −2.77814777577673496579588190349, −2.71488540180933286473424904123, −2.45743926949288468244277081776, −2.33068654746527686165677783329, −2.30883090086229616369517314445, −2.30617675650838253723635086167, −1.68738148471675665015981328022, −1.64587361621402961688956936471, −1.56372697768125446551306778271, −1.41903135605133277498661569868, −1.27309282765583189636570105611, −1.19105542465414532992862668459, −0.39143926442762275536399869671, −0.37173644232868418529930824677, −0.34400869778399379051097310846,
0.34400869778399379051097310846, 0.37173644232868418529930824677, 0.39143926442762275536399869671, 1.19105542465414532992862668459, 1.27309282765583189636570105611, 1.41903135605133277498661569868, 1.56372697768125446551306778271, 1.64587361621402961688956936471, 1.68738148471675665015981328022, 2.30617675650838253723635086167, 2.30883090086229616369517314445, 2.33068654746527686165677783329, 2.45743926949288468244277081776, 2.71488540180933286473424904123, 2.77814777577673496579588190349, 3.15188261646090351558425730707, 3.16697465462797393106678710151, 3.56428376748819566317412804000, 3.57495112561867474205829030122, 3.69165310339962767369560678964, 3.79803893725793442786446647294, 4.05960019480092712013542013190, 4.18499771242696888959210147481, 4.23071940256840709222236083897, 4.28796108798481282090230690788
Plot not available for L-functions of degree greater than 10.
