| L(s) = 1 | + 12·31-s − 8·37-s + 20·41-s − 8·43-s + 6·49-s + 12·53-s + 24·67-s − 8·71-s + 36·79-s − 32·83-s − 28·89-s − 40·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.15·31-s − 1.31·37-s + 3.12·41-s − 1.21·43-s + 6/7·49-s + 1.64·53-s + 2.93·67-s − 0.949·71-s + 4.05·79-s − 3.51·83-s − 2.96·89-s − 3.86·107-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.12260904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.12260904\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 11 T^{2} - 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 + 4 T + 51 T^{2} + 40 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 10 T + 87 T^{2} - 588 T^{3} + 87 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 + 4 T + 65 T^{2} + 216 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{6} \) |
| 71 | \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \) |
| 79 | \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( ( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( ( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.01351455437297958205452461789, −3.86079646097466538375952454262, −3.79506925113733542097764648936, −3.77334784152798513174311838691, −3.47008314354297202701059942539, −3.46300036666273292565506996775, −3.17011046640371357489119324534, −3.05553210248927935592162162498, −2.82671950975736014714025798236, −2.69588400804489880803599262782, −2.66311432027009156789929149104, −2.49393296027004046784090276515, −2.41854975370956662093350388954, −2.39720889123347830048113779802, −1.94409571360593540199707284351, −1.94138792493240170770397333560, −1.80447035017233137996324209056, −1.37183320681353736341220413611, −1.32171105544503658245726529245, −1.24702037831050993266105718060, −1.12840561205083510891657294757, −0.819293981875017042963549420639, −0.42129723274696451594403581946, −0.41761657947356253890185342682, −0.40628038905798949174375705398,
0.40628038905798949174375705398, 0.41761657947356253890185342682, 0.42129723274696451594403581946, 0.819293981875017042963549420639, 1.12840561205083510891657294757, 1.24702037831050993266105718060, 1.32171105544503658245726529245, 1.37183320681353736341220413611, 1.80447035017233137996324209056, 1.94138792493240170770397333560, 1.94409571360593540199707284351, 2.39720889123347830048113779802, 2.41854975370956662093350388954, 2.49393296027004046784090276515, 2.66311432027009156789929149104, 2.69588400804489880803599262782, 2.82671950975736014714025798236, 3.05553210248927935592162162498, 3.17011046640371357489119324534, 3.46300036666273292565506996775, 3.47008314354297202701059942539, 3.77334784152798513174311838691, 3.79506925113733542097764648936, 3.86079646097466538375952454262, 4.01351455437297958205452461789
Plot not available for L-functions of degree greater than 10.
