| L(s) = 1 | − 3·4-s − 12·11-s + 6·16-s − 4·19-s − 8·25-s − 4·29-s + 24·31-s − 4·41-s + 36·44-s + 8·49-s + 4·59-s + 20·61-s − 10·64-s + 16·71-s + 12·76-s − 56·79-s − 4·89-s + 24·100-s + 28·101-s − 48·109-s + 12·116-s + 18·121-s − 72·124-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 3.61·11-s + 3/2·16-s − 0.917·19-s − 8/5·25-s − 0.742·29-s + 4.31·31-s − 0.624·41-s + 5.42·44-s + 8/7·49-s + 0.520·59-s + 2.56·61-s − 5/4·64-s + 1.89·71-s + 1.37·76-s − 6.30·79-s − 0.423·89-s + 12/5·100-s + 2.78·101-s − 4.59·109-s + 1.11·116-s + 1.63·121-s − 6.46·124-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4238645349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4238645349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 8 T^{2} + 4 T^{3} + 8 p T^{4} + p^{3} T^{6} \) |
| 13 | \( ( 1 + T^{2} )^{3} \) |
| good | 7 | \( 1 - 8 T^{2} + 88 T^{4} - 734 T^{6} + 88 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 2 T + p T^{2} )^{6} \) |
| 17 | \( 1 + 752 T^{4} - 50 T^{6} + 752 p^{2} T^{8} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 13 T^{2} + 116 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 70 T^{2} + 2127 T^{4} - 47860 T^{6} + 2127 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 2 T + 43 T^{2} + 156 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 12 T + 113 T^{2} - 664 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 160 T^{2} + 11392 T^{4} - 506230 T^{6} + 11392 p^{2} T^{8} - 160 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 2 T + 43 T^{2} - 156 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 84 T^{2} + 5136 T^{4} - 202462 T^{6} + 5136 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 216 T^{2} + 21176 T^{4} - 1243838 T^{6} + 21176 p^{2} T^{8} - 216 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 250 T^{2} + 28167 T^{4} - 1878700 T^{6} + 28167 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 2 T + 133 T^{2} - 276 T^{3} + 133 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 10 T + 135 T^{2} - 1252 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 298 T^{2} + 41783 T^{4} - 3518604 T^{6} + 41783 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 - 8 T + 178 T^{2} - 936 T^{3} + 178 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 + 28 T + 453 T^{2} + 4744 T^{3} + 453 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 322 T^{2} + 51623 T^{4} - 5250876 T^{6} + 51623 p^{2} T^{8} - 322 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 2 T + 223 T^{2} + 396 T^{3} + 223 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 10 T^{2} + 4687 T^{4} - 11980 T^{6} + 4687 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.29158511588409829874106976146, −5.02008764388490064653119302938, −4.77999458888552860098192226859, −4.74749992766504637091983227286, −4.39337530530037830456033414902, −4.37557459772107776443062557125, −4.34648756712762066733214306443, −4.04350013553737172658457646786, −3.90216728695400336197125854639, −3.82747306314740883253021311495, −3.54281237195181493593853758740, −3.19936425922261964593591028157, −3.08300638830055572680234468864, −3.03828797902457109363099467705, −2.60841754072554068031559434368, −2.54927562617931576705112955131, −2.47188698417116416374993877768, −2.25221639044164979339684902519, −2.19872716927406850730121821739, −1.67350836674783665573746645895, −1.36293244205502631529179754541, −1.15958865512380511782563721364, −0.900262180629255667876591637477, −0.30615968849467756538030730029, −0.23797890330297328424662144559,
0.23797890330297328424662144559, 0.30615968849467756538030730029, 0.900262180629255667876591637477, 1.15958865512380511782563721364, 1.36293244205502631529179754541, 1.67350836674783665573746645895, 2.19872716927406850730121821739, 2.25221639044164979339684902519, 2.47188698417116416374993877768, 2.54927562617931576705112955131, 2.60841754072554068031559434368, 3.03828797902457109363099467705, 3.08300638830055572680234468864, 3.19936425922261964593591028157, 3.54281237195181493593853758740, 3.82747306314740883253021311495, 3.90216728695400336197125854639, 4.04350013553737172658457646786, 4.34648756712762066733214306443, 4.37557459772107776443062557125, 4.39337530530037830456033414902, 4.74749992766504637091983227286, 4.77999458888552860098192226859, 5.02008764388490064653119302938, 5.29158511588409829874106976146
Plot not available for L-functions of degree greater than 10.
