| L(s) = 1 | − 6·2-s + 21·4-s − 56·8-s − 12·9-s − 4·11-s + 126·16-s + 72·18-s + 24·22-s − 32·23-s + 6·25-s − 6·29-s − 252·32-s − 252·36-s − 20·37-s − 20·43-s − 84·44-s + 192·46-s − 36·50-s + 28·53-s + 36·58-s + 462·64-s − 24·67-s − 24·71-s + 672·72-s + 120·74-s − 8·79-s + 69·81-s + ⋯ |
| L(s) = 1 | − 4.24·2-s + 21/2·4-s − 19.7·8-s − 4·9-s − 1.20·11-s + 63/2·16-s + 16.9·18-s + 5.11·22-s − 6.67·23-s + 6/5·25-s − 1.11·29-s − 44.5·32-s − 42·36-s − 3.28·37-s − 3.04·43-s − 12.6·44-s + 28.3·46-s − 5.09·50-s + 3.84·53-s + 4.72·58-s + 57.7·64-s − 2.93·67-s − 2.84·71-s + 79.1·72-s + 13.9·74-s − 0.900·79-s + 23/3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{6} \) |
| 7 | \( 1 \) |
| 29 | \( ( 1 + T )^{6} \) |
| good | 3 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{3} \) |
| 5 | \( 1 - 6 T^{2} - 21 T^{4} + 332 T^{6} - 21 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 2 T + 9 T^{2} + 68 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 42 T^{2} + 987 T^{4} + 15852 T^{6} + 987 p^{2} T^{8} + 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 36 T^{2} + 63 p T^{4} + 18704 T^{6} + 63 p^{3} T^{8} + 36 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 44 T^{2} + 1323 T^{4} + 31576 T^{6} + 1323 p^{2} T^{8} + 44 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 16 T + 137 T^{2} + 792 T^{3} + 137 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 26 T^{2} + 61 p T^{4} - 23836 T^{6} + 61 p^{3} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 + 10 T + 39 T^{2} + 92 T^{3} + 39 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 156 T^{2} + 11823 T^{4} + 582848 T^{6} + 11823 p^{2} T^{8} + 156 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 10 T + 93 T^{2} + 692 T^{3} + 93 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 14 T^{2} + 1043 T^{4} + 183284 T^{6} + 1043 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 14 T + 155 T^{2} - 1236 T^{3} + 155 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 304 T^{2} + 40919 T^{4} + 3122104 T^{6} + 40919 p^{2} T^{8} + 304 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 + 254 T^{2} + 30087 T^{4} + 2229700 T^{6} + 30087 p^{2} T^{8} + 254 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 + 4 T + p T^{2} )^{6} \) |
| 71 | \( ( 1 + 12 T + 105 T^{2} + 1056 T^{3} + 105 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 220 T^{2} + 22015 T^{4} + 1638752 T^{6} + 22015 p^{2} T^{8} + 220 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 4 T + 193 T^{2} + 464 T^{3} + 193 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 232 T^{2} + 32279 T^{4} + 3301864 T^{6} + 32279 p^{2} T^{8} + 232 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 204 T^{2} + 26943 T^{4} + 2951552 T^{6} + 26943 p^{2} T^{8} + 204 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 340 T^{2} + 61999 T^{4} + 7376624 T^{6} + 61999 p^{2} T^{8} + 340 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.23349995403181158808846508431, −5.02888557285572214949486887636, −4.69042689392332505033868951061, −4.50324920183427532500792525706, −4.45104698575383724525111613028, −4.09380733226533891378073575589, −4.05439562917161984618557032450, −3.61316680534941185293643352845, −3.58717597726373302331172034397, −3.51353998166103385787850300413, −3.49149443016754521897278706996, −3.30958097112750094733663075989, −2.94328950736967876146924692180, −2.81102255067603518119379563090, −2.72096123997181310190532156504, −2.39818515534912050775541502373, −2.38183526413770336734782166320, −2.33260983902299690087688273628, −2.20731167696318776674831660043, −1.97384073207015165481187856345, −1.71952934886559228843554518499, −1.55701361836534896379972449401, −1.31778831357671094342583396991, −1.23271199009795493291317899199, −0.903968430235204345769499214012, 0, 0, 0, 0, 0, 0,
0.903968430235204345769499214012, 1.23271199009795493291317899199, 1.31778831357671094342583396991, 1.55701361836534896379972449401, 1.71952934886559228843554518499, 1.97384073207015165481187856345, 2.20731167696318776674831660043, 2.33260983902299690087688273628, 2.38183526413770336734782166320, 2.39818515534912050775541502373, 2.72096123997181310190532156504, 2.81102255067603518119379563090, 2.94328950736967876146924692180, 3.30958097112750094733663075989, 3.49149443016754521897278706996, 3.51353998166103385787850300413, 3.58717597726373302331172034397, 3.61316680534941185293643352845, 4.05439562917161984618557032450, 4.09380733226533891378073575589, 4.45104698575383724525111613028, 4.50324920183427532500792525706, 4.69042689392332505033868951061, 5.02888557285572214949486887636, 5.23349995403181158808846508431
Plot not available for L-functions of degree greater than 10.
