| L(s) = 1 | − 4·4-s − 6·9-s + 10·16-s − 12·19-s + 18·29-s + 8·31-s + 24·36-s + 14·41-s − 18·49-s + 4·59-s − 6·61-s − 23·64-s − 12·71-s + 48·76-s + 52·79-s + 10·81-s − 20·89-s + 26·101-s + 12·109-s − 72·116-s − 40·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s − 60·144-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2·9-s + 5/2·16-s − 2.75·19-s + 3.34·29-s + 1.43·31-s + 4·36-s + 2.18·41-s − 2.57·49-s + 0.520·59-s − 0.768·61-s − 2.87·64-s − 1.42·71-s + 5.50·76-s + 5.85·79-s + 10/9·81-s − 2.11·89-s + 2.58·101-s + 1.14·109-s − 6.68·116-s − 3.63·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.264569786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264569786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p^{2} T^{2} + 3 p T^{4} + 7 T^{6} + 3 p^{3} T^{8} + p^{6} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 + 2 p T^{2} + 26 T^{4} + 98 T^{6} + 26 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 + 18 T^{2} + 242 T^{4} + 1910 T^{6} + 242 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 20 T^{2} + 8 T^{3} + 20 p T^{4} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 + 67 T^{2} + 2118 T^{4} + 42943 T^{6} + 2118 p^{2} T^{8} + 67 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 6 T + 56 T^{2} + 218 T^{3} + 56 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 + 126 T^{2} + 6866 T^{4} + 206858 T^{6} + 6866 p^{2} T^{8} + 126 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 3 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 - 4 T + 53 T^{2} - 288 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 + 187 T^{2} + 15518 T^{4} + 737463 T^{6} + 15518 p^{2} T^{8} + 187 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 7 T + 94 T^{2} - 579 T^{3} + 94 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 + 178 T^{2} + 14258 T^{4} + 726702 T^{6} + 14258 p^{2} T^{8} + 178 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 + 46 T^{2} + 3407 T^{4} + 54276 T^{6} + 3407 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 + 147 T^{2} + 6923 T^{4} + 205346 T^{6} + 6923 p^{2} T^{8} + 147 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 122 T^{2} - 100 T^{3} + 122 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 3 T + 134 T^{2} + 251 T^{3} + 134 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 + 302 T^{2} + 43138 T^{4} + 3650498 T^{6} + 43138 p^{2} T^{8} + 302 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 6 T + 212 T^{2} + 826 T^{3} + 212 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 223 T^{2} + 31055 T^{4} + 2685330 T^{6} + 31055 p^{2} T^{8} + 223 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 26 T + 417 T^{2} - 4268 T^{3} + 417 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 222 T^{2} + 35303 T^{4} + 3305156 T^{6} + 35303 p^{2} T^{8} + 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 10 T + 110 T^{2} + 194 T^{3} + 110 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 + 302 T^{2} + 46858 T^{4} + 5028458 T^{6} + 46858 p^{2} T^{8} + 302 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.37802272702170716333166625957, −4.37800626209456910675083352982, −3.97996591294336267499799819353, −3.96686183135869126607090554665, −3.86600871743232241385164932024, −3.75557606372247443382727636303, −3.25018054997023861453610621300, −3.18813586794330586070947114441, −3.12194422943326834700673441747, −3.09159398251797629553193248040, −2.99906799868119078641194897988, −2.75110357565396219046321767923, −2.74670941997919037765028651247, −2.29165779823596722046264904973, −2.19597606349812059824651486196, −2.00823243963278197745698612367, −1.96415111120323327835945604441, −1.88622196251371329653174069856, −1.44183145116826615498407684104, −1.12408158942621849740862583341, −1.04774340187716176766708974778, −0.67617993828912182715233039440, −0.65068776833948751507647797009, −0.42144128874520658845038531278, −0.25990794276398897330170582100,
0.25990794276398897330170582100, 0.42144128874520658845038531278, 0.65068776833948751507647797009, 0.67617993828912182715233039440, 1.04774340187716176766708974778, 1.12408158942621849740862583341, 1.44183145116826615498407684104, 1.88622196251371329653174069856, 1.96415111120323327835945604441, 2.00823243963278197745698612367, 2.19597606349812059824651486196, 2.29165779823596722046264904973, 2.74670941997919037765028651247, 2.75110357565396219046321767923, 2.99906799868119078641194897988, 3.09159398251797629553193248040, 3.12194422943326834700673441747, 3.18813586794330586070947114441, 3.25018054997023861453610621300, 3.75557606372247443382727636303, 3.86600871743232241385164932024, 3.96686183135869126607090554665, 3.97996591294336267499799819353, 4.37800626209456910675083352982, 4.37802272702170716333166625957
Plot not available for L-functions of degree greater than 10.
