| L(s) = 1 | − 28·7-s − 27·9-s + 52·17-s − 328·23-s + 322·25-s + 636·31-s + 236·41-s + 408·47-s − 310·49-s + 756·63-s + 1.70e3·71-s + 956·73-s + 44·79-s + 486·81-s − 220·89-s − 2.44e3·97-s + 1.32e3·103-s + 5.14e3·113-s − 1.45e3·119-s + 2.35e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.40e3·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s + 0.741·17-s − 2.97·23-s + 2.57·25-s + 3.68·31-s + 0.898·41-s + 1.26·47-s − 0.903·49-s + 1.51·63-s + 2.84·71-s + 1.53·73-s + 0.0626·79-s + 2/3·81-s − 0.262·89-s − 2.55·97-s + 1.26·103-s + 4.28·113-s − 1.12·119-s + 1.76·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.741·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.579223724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.579223724\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| good | 5 | \( 1 - 322 T^{2} + 49351 T^{4} - 6170684 T^{6} + 49351 p^{6} T^{8} - 322 p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( ( 1 + 2 p T + 449 T^{2} + 3788 T^{3} + 449 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 11 | \( 1 - 214 p T^{2} + 37343 p^{2} T^{4} - 5987722076 T^{6} + 37343 p^{8} T^{8} - 214 p^{13} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 8270 T^{2} + 36264983 T^{4} - 97600232804 T^{6} + 36264983 p^{6} T^{8} - 8270 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( ( 1 - 26 T + 3615 T^{2} + 222100 T^{3} + 3615 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 19 | \( 1 - 18194 T^{2} + 9648173 p T^{4} - 1372700323292 T^{6} + 9648173 p^{7} T^{8} - 18194 p^{12} T^{10} + p^{18} T^{12} \) |
| 23 | \( ( 1 + 164 T + 42885 T^{2} + 4036280 T^{3} + 42885 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 29 | \( 1 - 123986 T^{2} + 6779237687 T^{4} - 212188653261788 T^{6} + 6779237687 p^{6} T^{8} - 123986 p^{12} T^{10} + p^{18} T^{12} \) |
| 31 | \( ( 1 - 318 T + 93849 T^{2} - 15197452 T^{3} + 93849 p^{3} T^{4} - 318 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 124142 T^{2} + 4233590471 T^{4} - 51775900405988 T^{6} + 4233590471 p^{6} T^{8} - 124142 p^{12} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 118 T + 89463 T^{2} + 3720620 T^{3} + 89463 p^{3} T^{4} - 118 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 247490 T^{2} + 32846327015 T^{4} - 3025278566260412 T^{6} + 32846327015 p^{6} T^{8} - 247490 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( ( 1 - 204 T + 283677 T^{2} - 40395048 T^{3} + 283677 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 53 | \( 1 - 292802 T^{2} + 260255467 p T^{4} + 2699981198933572 T^{6} + 260255467 p^{7} T^{8} - 292802 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( 1 - 1093858 T^{2} + 524838290887 T^{4} - 140555838506313212 T^{6} + 524838290887 p^{6} T^{8} - 1093858 p^{12} T^{10} + p^{18} T^{12} \) |
| 61 | \( 1 - 459870 T^{2} + 162687674679 T^{4} - 38865284671151684 T^{6} + 162687674679 p^{6} T^{8} - 459870 p^{12} T^{10} + p^{18} T^{12} \) |
| 67 | \( 1 - 750066 T^{2} + 226548162807 T^{4} - 53949461413257884 T^{6} + 226548162807 p^{6} T^{8} - 750066 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 - 12 p T + 1006773 T^{2} - 524795352 T^{3} + 1006773 p^{3} T^{4} - 12 p^{7} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( ( 1 - 478 T + 911095 T^{2} - 251066948 T^{3} + 911095 p^{3} T^{4} - 478 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 79 | \( ( 1 - 22 T + 1407593 T^{2} - 13791100 T^{3} + 1407593 p^{3} T^{4} - 22 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 2910274 T^{2} + 3775777045015 T^{4} - 2787348361783974908 T^{6} + 3775777045015 p^{6} T^{8} - 2910274 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 + 110 T + 2073543 T^{2} + 156516836 T^{3} + 2073543 p^{3} T^{4} + 110 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( ( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
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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
−7.18925671613010752966546073129, −6.96471856755003995010037164090, −6.92620319782875688718453191580, −6.85526711233969870412937232531, −6.28294529778078933220065088040, −6.20005229814512931253778175358, −6.05078342732781860751208086385, −5.94023095870860661389826426389, −5.87550081260865214497884497299, −5.12778777022101366021161096721, −5.10358281834891909894618662917, −4.94026200446015797201893645864, −4.59911670497521363428999385902, −4.22645757138495605512171968302, −4.06619187001677496390527624704, −3.68621033796480113681684864366, −3.40769092306494803031187260459, −3.12210026545588998535672609522, −2.84130777782109793203754866122, −2.68575984375190588330567942750, −2.21646132052290214849965656939, −1.96172237674734729345538275186, −1.09999676689116059091841406697, −0.74124780690162145530146580513, −0.42649129458420919847436324333,
0.42649129458420919847436324333, 0.74124780690162145530146580513, 1.09999676689116059091841406697, 1.96172237674734729345538275186, 2.21646132052290214849965656939, 2.68575984375190588330567942750, 2.84130777782109793203754866122, 3.12210026545588998535672609522, 3.40769092306494803031187260459, 3.68621033796480113681684864366, 4.06619187001677496390527624704, 4.22645757138495605512171968302, 4.59911670497521363428999385902, 4.94026200446015797201893645864, 5.10358281834891909894618662917, 5.12778777022101366021161096721, 5.87550081260865214497884497299, 5.94023095870860661389826426389, 6.05078342732781860751208086385, 6.20005229814512931253778175358, 6.28294529778078933220065088040, 6.85526711233969870412937232531, 6.92620319782875688718453191580, 6.96471856755003995010037164090, 7.18925671613010752966546073129
Plot not available for L-functions of degree greater than 10.
