| L(s) = 1 | + 6·3-s + 11·9-s + 30·11-s + 30·17-s − 78·19-s + 121·25-s + 2·27-s + 180·33-s + 116·41-s + 100·43-s + 180·51-s − 468·57-s + 110·59-s + 434·67-s + 102·73-s + 726·75-s − 69·81-s − 268·83-s + 214·89-s + 76·97-s + 330·99-s + 102·107-s − 340·113-s − 33·121-s + 696·123-s + 127-s + 600·129-s + ⋯ |
| L(s) = 1 | + 2·3-s + 11/9·9-s + 2.72·11-s + 1.76·17-s − 4.10·19-s + 4.83·25-s + 2/27·27-s + 5.45·33-s + 2.82·41-s + 2.32·43-s + 3.52·51-s − 8.21·57-s + 1.86·59-s + 6.47·67-s + 1.39·73-s + 9.67·75-s − 0.851·81-s − 3.22·83-s + 2.40·89-s + 0.783·97-s + 10/3·99-s + 0.953·107-s − 3.00·113-s − 0.272·121-s + 5.65·123-s + 0.00787·127-s + 4.65·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(27.47854050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.47854050\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 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 | \( ( 1 - 38 T + 25191 T^{2} - 597344 T^{3} + 25191 p^{2} T^{4} - 38 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
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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
−4.74883737282121118537664785664, −4.53875827817141880757715261805, −4.13226466579646377069057945924, −4.10469751386146326803052306177, −4.08765375232782485773894992645, −3.93535027184764630290082169084, −3.81394019550361506455689599206, −3.78845760708855034624380674393, −3.39024991748378892324776435557, −3.27165030779780969772924121247, −3.11483756763658451078048963641, −2.91755405259287685301112606537, −2.72790279430548673469644657629, −2.42836855708427262242828649901, −2.41698511169193025268244464073, −2.33358371217836724220002132183, −2.20570783616331071474962106851, −2.03693785261531938397701576922, −1.56602600211197509673817923541, −1.21149782416195337261483755221, −1.14793694236700536597207148640, −1.06954913281221397320710555831, −0.791794856812645715702525916067, −0.71727620885967321015820546475, −0.23440784154300963452978010579,
0.23440784154300963452978010579, 0.71727620885967321015820546475, 0.791794856812645715702525916067, 1.06954913281221397320710555831, 1.14793694236700536597207148640, 1.21149782416195337261483755221, 1.56602600211197509673817923541, 2.03693785261531938397701576922, 2.20570783616331071474962106851, 2.33358371217836724220002132183, 2.41698511169193025268244464073, 2.42836855708427262242828649901, 2.72790279430548673469644657629, 2.91755405259287685301112606537, 3.11483756763658451078048963641, 3.27165030779780969772924121247, 3.39024991748378892324776435557, 3.78845760708855034624380674393, 3.81394019550361506455689599206, 3.93535027184764630290082169084, 4.08765375232782485773894992645, 4.10469751386146326803052306177, 4.13226466579646377069057945924, 4.53875827817141880757715261805, 4.74883737282121118537664785664
Plot not available for L-functions of degree greater than 10.
