| L(s) = 1 | + 2·4-s − 6·5-s + 8·9-s + 16-s − 12·20-s + 13·25-s + 8·31-s + 16·36-s − 48·45-s + 40·49-s − 32·59-s + 16·64-s + 112·71-s − 6·80-s + 12·81-s − 108·89-s + 26·100-s + 16·124-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 8·144-s + 149-s + 151-s − 48·155-s + ⋯ |
| L(s) = 1 | + 4-s − 2.68·5-s + 8/3·9-s + 1/4·16-s − 2.68·20-s + 13/5·25-s + 1.43·31-s + 8/3·36-s − 7.15·45-s + 40/7·49-s − 4.16·59-s + 2·64-s + 13.2·71-s − 0.670·80-s + 4/3·81-s − 11.4·89-s + 13/5·100-s + 1.43·124-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-s + 0.0819·149-s + 0.0813·151-s − 3.85·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 11^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.004098216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004098216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( ( 1 + 3 T + 7 T^{2} + 18 T^{3} + 7 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 \) |
| good | 2 | \( ( 1 - T^{2} + T^{4} - 9 T^{6} + p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 3 | \( ( 1 - 4 T^{2} + 2 p^{2} T^{4} - 2 p^{3} T^{6} + 2 p^{4} T^{8} - 4 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( ( 1 - 20 T^{2} + 250 T^{4} - 2118 T^{6} + 250 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 41 T^{2} + 883 T^{4} - 13386 T^{6} + 883 p^{2} T^{8} - 41 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 5 p T^{2} + 3235 T^{4} - 70638 T^{6} + 3235 p^{2} T^{8} - 5 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 42 T^{2} + 1275 T^{4} + 27956 T^{6} + 1275 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 58 T^{2} + 2651 T^{4} - 67492 T^{6} + 2651 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 + 99 T^{2} + 5139 T^{4} + 176434 T^{6} + 5139 p^{2} T^{8} + 99 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 2 T + 55 T^{2} - 192 T^{3} + 55 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 37 | \( ( 1 - 171 T^{2} + 13491 T^{4} - 630146 T^{6} + 13491 p^{2} T^{8} - 171 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 + 129 T^{2} + 7386 T^{4} + 307609 T^{6} + 7386 p^{2} T^{8} + 129 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 128 T^{2} + 6634 T^{4} - 255762 T^{6} + 6634 p^{2} T^{8} - 128 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 268 T^{2} + 30554 T^{4} - 1895662 T^{6} + 30554 p^{2} T^{8} - 268 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 271 T^{2} + 32411 T^{4} - 2210806 T^{6} + 32411 p^{2} T^{8} - 271 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 8 T + 127 T^{2} + 968 T^{3} + 127 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 61 | \( ( 1 + 276 T^{2} + 35664 T^{4} + 2749250 T^{6} + 35664 p^{2} T^{8} + 276 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 228 T^{2} + 28578 T^{4} - 2344790 T^{6} + 28578 p^{2} T^{8} - 228 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 28 T + 403 T^{2} - 3928 T^{3} + 403 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 73 | \( ( 1 - 2 p T^{2} + 16687 T^{4} - 1215900 T^{6} + 16687 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 174 T^{2} + 14943 T^{4} + 1064612 T^{6} + 14943 p^{2} T^{8} + 174 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 202 T^{2} + 30979 T^{4} - 2911524 T^{6} + 30979 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 9 T + p T^{2} )^{12} \) |
| 97 | \( ( 1 - 63 T^{2} + 26883 T^{4} - 1111430 T^{6} + 26883 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.57099210397297784589929995326, −3.55615276908537337301604075137, −3.30659519405529082617728026099, −3.08535281658664906663450322100, −3.05806487072659030368895188540, −3.02450550710207103271622898756, −2.83980059568305158440234054365, −2.82079919852587197901969532535, −2.58991873612344909437720060403, −2.55729730086713467097000024898, −2.54040354733969044511358864049, −2.32432239475568633857193564091, −2.20032537922011504214848344880, −2.00965761417593815742162042937, −1.89418275326684755762421302192, −1.84783548590967637207233878773, −1.73106395257224948104542806078, −1.58905875222824457559931772424, −1.26337410015493563998134626053, −1.17395357696733623241416390947, −1.04395953203956498964817090481, −0.799726785967387345463535131481, −0.77332843046391827927371427543, −0.63050858535318676273797077770, −0.13414995616613161244988953462,
0.13414995616613161244988953462, 0.63050858535318676273797077770, 0.77332843046391827927371427543, 0.799726785967387345463535131481, 1.04395953203956498964817090481, 1.17395357696733623241416390947, 1.26337410015493563998134626053, 1.58905875222824457559931772424, 1.73106395257224948104542806078, 1.84783548590967637207233878773, 1.89418275326684755762421302192, 2.00965761417593815742162042937, 2.20032537922011504214848344880, 2.32432239475568633857193564091, 2.54040354733969044511358864049, 2.55729730086713467097000024898, 2.58991873612344909437720060403, 2.82079919852587197901969532535, 2.83980059568305158440234054365, 3.02450550710207103271622898756, 3.05806487072659030368895188540, 3.08535281658664906663450322100, 3.30659519405529082617728026099, 3.55615276908537337301604075137, 3.57099210397297784589929995326
Plot not available for L-functions of degree greater than 10.
