| L(s) = 1 | − 3-s − 3·5-s + 5·7-s + 3·9-s + 2·11-s + 3·15-s + 4·17-s − 8·19-s − 5·21-s + 7·23-s + 3·25-s − 2·27-s + 7·29-s + 16·31-s − 2·33-s − 15·35-s + 4·37-s + 41-s − 2·43-s − 9·45-s + 13·47-s + 18·49-s − 4·51-s − 20·53-s − 6·55-s + 8·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.88·7-s + 9-s + 0.603·11-s + 0.774·15-s + 0.970·17-s − 1.83·19-s − 1.09·21-s + 1.45·23-s + 3/5·25-s − 0.384·27-s + 1.29·29-s + 2.87·31-s − 0.348·33-s − 2.53·35-s + 0.657·37-s + 0.156·41-s − 0.304·43-s − 1.34·45-s + 1.89·47-s + 18/7·49-s − 0.560·51-s − 2.74·53-s − 0.809·55-s + 1.05·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.076257995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.076257995\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T - 2 T^{2} - p T^{3} - 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | \( ( 1 + T + T^{2} )^{3} \) |
| good | 7 | \( 1 - 5 T + p T^{2} + 32 T^{3} - 107 T^{4} - 83 T^{5} + 914 T^{6} - 83 p T^{7} - 107 p^{2} T^{8} + 32 p^{3} T^{9} + p^{5} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \) |
| 11 | \( 1 - 2 T - 21 T^{2} + 14 T^{3} + 26 p T^{4} + 58 T^{5} - 3673 T^{6} + 58 p T^{7} + 26 p^{3} T^{8} + 14 p^{3} T^{9} - 21 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 13 | \( 1 - 15 T^{2} + 72 T^{3} + 30 T^{4} - 540 T^{5} + 2765 T^{6} - 540 p T^{7} + 30 p^{2} T^{8} + 72 p^{3} T^{9} - 15 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 - 2 T + 15 T^{2} + 40 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( ( 1 + 4 T + 53 T^{2} + 148 T^{3} + 53 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 7 T + 31 T^{2} - 88 T^{3} - 223 T^{4} + 1751 T^{5} - 8678 T^{6} + 1751 p T^{7} - 223 p^{2} T^{8} - 88 p^{3} T^{9} + 31 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( 1 - 7 T - 33 T^{2} + 184 T^{3} + 1693 T^{4} - 3673 T^{5} - 43786 T^{6} - 3673 p T^{7} + 1693 p^{2} T^{8} + 184 p^{3} T^{9} - 33 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12} \) |
| 31 | \( 1 - 16 T + 87 T^{2} - 504 T^{3} + 5722 T^{4} - 33628 T^{5} + 138175 T^{6} - 33628 p T^{7} + 5722 p^{2} T^{8} - 504 p^{3} T^{9} + 87 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 2 T + 103 T^{2} - 136 T^{3} + 103 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 - T - 113 T^{2} + 56 T^{3} + 8237 T^{4} - 2023 T^{5} - 389450 T^{6} - 2023 p T^{7} + 8237 p^{2} T^{8} + 56 p^{3} T^{9} - 113 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \) |
| 43 | \( 1 + 2 T - 89 T^{2} + 58 T^{3} + 4522 T^{4} - 6346 T^{5} - 216193 T^{6} - 6346 p T^{7} + 4522 p^{2} T^{8} + 58 p^{3} T^{9} - 89 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \) |
| 47 | \( 1 - 13 T + 113 T^{2} - 530 T^{3} - 783 T^{4} + 34171 T^{5} - 303266 T^{6} + 34171 p T^{7} - 783 p^{2} T^{8} - 530 p^{3} T^{9} + 113 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 10 T + 171 T^{2} + 1036 T^{3} + 171 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 6 T - 81 T^{2} + 858 T^{3} + 2106 T^{4} - 29310 T^{5} + 86191 T^{6} - 29310 p T^{7} + 2106 p^{2} T^{8} + 858 p^{3} T^{9} - 81 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 11 T - 21 T^{2} + 84 T^{3} + 2449 T^{4} + 32767 T^{5} - 548954 T^{6} + 32767 p T^{7} + 2449 p^{2} T^{8} + 84 p^{3} T^{9} - 21 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + T - p T^{2} - 426 T^{3} - 179 T^{4} + 12173 T^{5} + 348238 T^{6} + 12173 p T^{7} - 179 p^{2} T^{8} - 426 p^{3} T^{9} - p^{5} T^{10} + p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 14 T + 193 T^{2} + 1952 T^{3} + 193 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 16 T + 3 p T^{2} - 1952 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 79 | \( 1 - 6 T - 117 T^{2} + 1186 T^{3} + 5010 T^{4} - 53358 T^{5} + 18795 T^{6} - 53358 p T^{7} + 5010 p^{2} T^{8} + 1186 p^{3} T^{9} - 117 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( 1 - 21 T + 63 T^{2} - 480 T^{3} + 33201 T^{4} - 236523 T^{5} + 151594 T^{6} - 236523 p T^{7} + 33201 p^{2} T^{8} - 480 p^{3} T^{9} + 63 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 11 T + p T^{2} )^{6} \) |
| 97 | \( 1 + 30 T + 453 T^{2} + 3962 T^{3} + 14730 T^{4} - 207642 T^{5} - 3708051 T^{6} - 207642 p T^{7} + 14730 p^{2} T^{8} + 3962 p^{3} T^{9} + 453 p^{4} T^{10} + 30 p^{5} T^{11} + p^{6} T^{12} \) |
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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
−6.40309837379931247392603756524, −5.91370454537385295608148231578, −5.74611534519411079612301020298, −5.70118648257996675860530200226, −5.59380555724354491844693425660, −5.31379447278667365148163704674, −5.00812274795593065185813647777, −4.68550884616174101006525334097, −4.67261369948393985556976389801, −4.43042167919994180223881376668, −4.42425081229198315781460175294, −4.41094697396850515100216317704, −4.01325598142236583907545408187, −3.89244387469709192370114444810, −3.50459121958554878090589360108, −3.20806757218459966568254690268, −3.11738144054325789498189335654, −2.87188052615094909976938975548, −2.36196969136163863145051512041, −2.33976794203741960542706778660, −2.03622774309509672050065634636, −1.44552161298896703227627031468, −1.32422863162694981497057729938, −0.919666056207958940391486868737, −0.68728305420562628566961483884,
0.68728305420562628566961483884, 0.919666056207958940391486868737, 1.32422863162694981497057729938, 1.44552161298896703227627031468, 2.03622774309509672050065634636, 2.33976794203741960542706778660, 2.36196969136163863145051512041, 2.87188052615094909976938975548, 3.11738144054325789498189335654, 3.20806757218459966568254690268, 3.50459121958554878090589360108, 3.89244387469709192370114444810, 4.01325598142236583907545408187, 4.41094697396850515100216317704, 4.42425081229198315781460175294, 4.43042167919994180223881376668, 4.67261369948393985556976389801, 4.68550884616174101006525334097, 5.00812274795593065185813647777, 5.31379447278667365148163704674, 5.59380555724354491844693425660, 5.70118648257996675860530200226, 5.74611534519411079612301020298, 5.91370454537385295608148231578, 6.40309837379931247392603756524
Plot not available for L-functions of degree greater than 10.
