| L(s) = 1 | + 4·4-s − 10·9-s − 8·13-s + 8·16-s + 8·17-s + 6·23-s + 19·25-s + 4·27-s − 14·29-s − 40·36-s − 26·43-s − 32·52-s + 2·53-s − 28·61-s + 12·64-s + 32·68-s + 26·79-s + 45·81-s + 24·92-s + 76·100-s + 16·101-s + 48·103-s − 16·107-s + 16·108-s − 10·113-s − 56·116-s + 80·117-s + ⋯ |
| L(s) = 1 | + 2·4-s − 3.33·9-s − 2.21·13-s + 2·16-s + 1.94·17-s + 1.25·23-s + 19/5·25-s + 0.769·27-s − 2.59·29-s − 6.66·36-s − 3.96·43-s − 4.43·52-s + 0.274·53-s − 3.58·61-s + 3/2·64-s + 3.88·68-s + 2.92·79-s + 5·81-s + 2.50·92-s + 38/5·100-s + 1.59·101-s + 4.72·103-s − 1.54·107-s + 1.53·108-s − 0.940·113-s − 5.19·116-s + 7.39·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.765323815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765323815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 8 T + 7 T^{2} - 64 T^{3} + 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 2 | \( 1 - p^{2} T^{2} + p^{3} T^{4} - 3 p^{2} T^{6} + p^{5} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 3 | \( ( 1 + 5 T^{2} - 2 T^{3} + 5 p T^{4} + p^{3} T^{6} )^{2} \) |
| 5 | \( 1 - 19 T^{2} + 162 T^{4} - 919 T^{6} + 162 p^{2} T^{8} - 19 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 38 T^{2} + 747 T^{4} - 9800 T^{6} + 747 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 4 T + 43 T^{2} - 6 p T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 + 5 T^{2} + 238 T^{4} + 10877 T^{6} + 238 p^{2} T^{8} + 5 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 3 T + 44 T^{2} - 59 T^{3} + 44 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 7 T + 66 T^{2} + 411 T^{3} + 66 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 103 T^{2} + 5914 T^{4} - 224059 T^{6} + 5914 p^{2} T^{8} - 103 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 114 T^{2} + 6171 T^{4} - 242912 T^{6} + 6171 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 138 T^{2} + 10367 T^{4} - 522380 T^{6} + 10367 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 13 T + 164 T^{2} + 1101 T^{3} + 164 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 131 T^{2} + 10566 T^{4} - 603323 T^{6} + 10566 p^{2} T^{8} - 131 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - T + 150 T^{2} - 93 T^{3} + 150 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 286 T^{2} + 36983 T^{4} - 2780916 T^{6} + 36983 p^{2} T^{8} - 286 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 + 14 T + 211 T^{2} + 1556 T^{3} + 211 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 322 T^{2} + 47303 T^{4} - 4048956 T^{6} + 47303 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 122 T^{2} + 17115 T^{4} - 1124048 T^{6} + 17115 p^{2} T^{8} - 122 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 175 T^{2} + 10862 T^{4} - 497775 T^{6} + 10862 p^{2} T^{8} - 175 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 13 T + 200 T^{2} - 1869 T^{3} + 200 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 271 T^{2} + 41066 T^{4} - 4200123 T^{6} + 41066 p^{2} T^{8} - 271 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 415 T^{2} + 80838 T^{4} - 9173143 T^{6} + 80838 p^{2} T^{8} - 415 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 7 T^{2} + 14150 T^{4} + 326769 T^{6} + 14150 p^{2} T^{8} - 7 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.79624890033987750436802415283, −5.42399883371316029703684043398, −5.36306859188048973012810299913, −5.24155405214017147076790293919, −4.92744874377888851914935533350, −4.83993584534087142284875673239, −4.81931478374544776816971549563, −4.73437413379863539094759331377, −4.46799073174058105414722910058, −3.84550983473859407535910963826, −3.77321305723303121343986787842, −3.29020351358403761912655909648, −3.27817609460783243359897119902, −3.23684119880560077967834410851, −3.20789029472901558456484565075, −2.90942201973067085614059673007, −2.73634963356593528533385365846, −2.51940384551542578828819452204, −2.31406729583108467397504705361, −1.99214352820529198237866958438, −1.92332482781661870489225873009, −1.48151181388572481382849375305, −1.23055183737234014914427762653, −0.68473158128734591095679135972, −0.35376315368697938620362363447,
0.35376315368697938620362363447, 0.68473158128734591095679135972, 1.23055183737234014914427762653, 1.48151181388572481382849375305, 1.92332482781661870489225873009, 1.99214352820529198237866958438, 2.31406729583108467397504705361, 2.51940384551542578828819452204, 2.73634963356593528533385365846, 2.90942201973067085614059673007, 3.20789029472901558456484565075, 3.23684119880560077967834410851, 3.27817609460783243359897119902, 3.29020351358403761912655909648, 3.77321305723303121343986787842, 3.84550983473859407535910963826, 4.46799073174058105414722910058, 4.73437413379863539094759331377, 4.81931478374544776816971549563, 4.83993584534087142284875673239, 4.92744874377888851914935533350, 5.24155405214017147076790293919, 5.36306859188048973012810299913, 5.42399883371316029703684043398, 5.79624890033987750436802415283
Plot not available for L-functions of degree greater than 10.
