| L(s) = 1 | − 2·2-s + 6·3-s + 4·4-s − 12·6-s − 4·8-s + 11·9-s − 30·11-s + 24·12-s − 30·17-s − 22·18-s − 78·19-s + 60·22-s − 24·24-s + 121·25-s + 2·27-s − 16·32-s − 180·33-s + 60·34-s + 44·36-s + 156·38-s − 116·41-s − 100·43-s − 120·44-s − 242·50-s − 180·51-s − 4·54-s − 468·57-s + ⋯ |
| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 1/2·8-s + 11/9·9-s − 2.72·11-s + 2·12-s − 1.76·17-s − 1.22·18-s − 4.10·19-s + 2.72·22-s − 24-s + 4.83·25-s + 2/27·27-s − 1/2·32-s − 5.45·33-s + 1.76·34-s + 11/9·36-s + 4.10·38-s − 2.82·41-s − 2.32·43-s − 2.72·44-s − 4.83·50-s − 3.52·51-s − 0.0740·54-s − 8.21·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1084962583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1084962583\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T - p^{2} T^{3} + p^{5} T^{5} + p^{6} T^{6} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - p T + 8 T^{2} - 19 T^{3} + 8 p^{2} T^{4} - p^{5} T^{5} + p^{6} T^{6} )^{2} \) |
| 5 | \( 1 - 121 T^{2} + 6662 T^{4} - 212757 T^{6} + 6662 p^{4} T^{8} - 121 p^{8} T^{10} + p^{12} T^{12} \) |
| 11 | \( ( 1 + 15 T + 354 T^{2} + 3117 T^{3} + 354 p^{2} T^{4} + 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 60895 p^{4} T^{8} - 86 p^{8} T^{10} + p^{12} T^{12} \) |
| 17 | \( ( 1 + 15 T + 890 T^{2} + 8635 T^{3} + 890 p^{2} T^{4} + 15 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 19 | \( ( 1 + 39 T + 1370 T^{2} + 28697 T^{3} + 1370 p^{2} T^{4} + 39 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( 1 - 693 T^{2} + 495830 T^{4} - 329634845 T^{6} + 495830 p^{4} T^{8} - 693 p^{8} T^{10} + p^{12} T^{12} \) |
| 29 | \( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 6471151 p^{4} T^{8} - 3662 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( 1 - 3561 T^{2} + 6841842 T^{4} - 8041403549 T^{6} + 6841842 p^{4} T^{8} - 3561 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 5785 T^{2} + 15576398 T^{4} - 26031024189 T^{6} + 15576398 p^{4} T^{8} - 5785 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( ( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 3139 p^{2} T^{4} + 58 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 43 | \( ( 1 + 50 T + 4047 T^{2} + 107900 T^{3} + 4047 p^{2} T^{4} + 50 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( 1 - 3905 T^{2} + 8711938 T^{4} - 15970210469 T^{6} + 8711938 p^{4} T^{8} - 3905 p^{8} T^{10} + p^{12} T^{12} \) |
| 53 | \( 1 - 10561 T^{2} + 51800990 T^{4} - 168592871253 T^{6} + 51800990 p^{4} T^{8} - 10561 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( ( 1 - 55 T + 10392 T^{2} - 376351 T^{3} + 10392 p^{2} T^{4} - 55 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 61 | \( 1 - 9201 T^{2} + 27716990 T^{4} - 50012187845 T^{6} + 27716990 p^{4} T^{8} - 9201 p^{8} T^{10} + p^{12} T^{12} \) |
| 67 | \( ( 1 + 217 T + 29036 T^{2} + 2316921 T^{3} + 29036 p^{2} T^{4} + 217 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 71 | \( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 245626031 p^{4} T^{8} - 23062 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 + 51 T + 11766 T^{2} + 589863 T^{3} + 11766 p^{2} T^{4} + 51 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 16693 T^{2} + 129404966 T^{4} - 759834693213 T^{6} + 129404966 p^{4} T^{8} - 16693 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( ( 1 + 134 T + 22583 T^{2} + 1661172 T^{3} + 22583 p^{2} T^{4} + 134 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( ( 1 + 107 T + 10054 T^{2} + 786431 T^{3} + 10054 p^{2} T^{4} + 107 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 97 | \( ( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 25191 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.98261420426425792610906216568, −5.69689983202779952657147971050, −5.35403050255698098666354561228, −5.32689076700697679938481067925, −5.19511833702217177274439181236, −5.04082492088369032554833452244, −4.68182728085364477816686913588, −4.43489068523978594202845162014, −4.31967663035974375504380985143, −4.30826978214399022079721021148, −4.15091812569787561524996026982, −3.69398257583641894599821692577, −3.29539842760384300629506882072, −3.07075131264806299059779966798, −2.91807142215685703843322741208, −2.87931533254264251857657116691, −2.58999022847008148304349004239, −2.58886572035820390631975821054, −2.42375573946534933159980038622, −1.72296315673245442359091098279, −1.71455932388211848167665923543, −1.67118662276838657174493412630, −1.21816801581952445725892906328, −0.19171284642439009926403476298, −0.13071267130142097844542605164,
0.13071267130142097844542605164, 0.19171284642439009926403476298, 1.21816801581952445725892906328, 1.67118662276838657174493412630, 1.71455932388211848167665923543, 1.72296315673245442359091098279, 2.42375573946534933159980038622, 2.58886572035820390631975821054, 2.58999022847008148304349004239, 2.87931533254264251857657116691, 2.91807142215685703843322741208, 3.07075131264806299059779966798, 3.29539842760384300629506882072, 3.69398257583641894599821692577, 4.15091812569787561524996026982, 4.30826978214399022079721021148, 4.31967663035974375504380985143, 4.43489068523978594202845162014, 4.68182728085364477816686913588, 5.04082492088369032554833452244, 5.19511833702217177274439181236, 5.32689076700697679938481067925, 5.35403050255698098666354561228, 5.69689983202779952657147971050, 5.98261420426425792610906216568
Plot not available for L-functions of degree greater than 10.
