| L(s) = 1 | − 6·3-s + 21·9-s − 56·27-s − 24·29-s + 42·43-s − 3·49-s − 12·53-s + 6·61-s + 18·79-s + 126·81-s + 144·87-s + 24·101-s − 6·103-s − 24·107-s + 84·113-s + 18·121-s + 127-s − 252·129-s + 131-s + 137-s + 139-s + 18·147-s + 149-s + 151-s + 157-s + 72·159-s + 163-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s − 10.7·27-s − 4.45·29-s + 6.40·43-s − 3/7·49-s − 1.64·53-s + 0.768·61-s + 2.02·79-s + 14·81-s + 15.4·87-s + 2.38·101-s − 0.591·103-s − 2.32·107-s + 7.90·113-s + 1.63·121-s + 0.0887·127-s − 22.1·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.48·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 5.70·159-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3290922245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3290922245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{6} \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 246 T^{6} + p^{6} T^{12} \) |
| 7 | \( 1 + 3 T^{2} + 15 T^{4} + 106 T^{6} + 15 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 18 T^{2} + 279 T^{4} - 4380 T^{6} + 279 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 30 T^{2} - 16 T^{3} + 30 p T^{4} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 6 T^{2} + 999 T^{4} - 4084 T^{6} + 999 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 45 T^{2} - 8 T^{3} + 45 p T^{4} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 + 12 T + 120 T^{2} + 702 T^{3} + 120 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 - 153 T^{2} + 10563 T^{4} - 420166 T^{6} + 10563 p^{2} T^{8} - 153 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( 1 - 168 T^{2} + 12840 T^{4} - 591158 T^{6} + 12840 p^{2} T^{8} - 168 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( 1 - 132 T^{2} + 9768 T^{4} - 470698 T^{6} + 9768 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 - 21 T + 255 T^{2} - 2018 T^{3} + 255 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 186 T^{2} + 16239 T^{4} - 911788 T^{6} + 16239 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 + 6 T + 36 T^{2} + 428 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 - 174 T^{2} + 15303 T^{4} - 1004100 T^{6} + 15303 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 3 T + 138 T^{2} - 199 T^{3} + 138 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 117 T^{2} + 11943 T^{4} - 876886 T^{6} + 11943 p^{2} T^{8} - 117 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( 1 - 330 T^{2} + 49887 T^{4} - 4471116 T^{6} + 49887 p^{2} T^{8} - 330 p^{4} T^{10} + p^{6} T^{12} \) |
| 73 | \( 1 - 291 T^{2} + 41142 T^{4} - 3654727 T^{6} + 41142 p^{2} T^{8} - 291 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 9 T + 117 T^{2} - 1438 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 150 T^{2} + 285 p T^{4} - 2019540 T^{6} + 285 p^{3} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 - 222 T^{2} + 22863 T^{4} - 1960324 T^{6} + 22863 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 - 405 T^{2} + 79971 T^{4} - 9662734 T^{6} + 79971 p^{2} T^{8} - 405 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−4.41718084189637908177034038363, −4.27831096655621500687976491923, −4.14192347461549325542775835977, −4.03796807114561316068123306031, −3.74898868184369707262172114265, −3.74036745745582756835684696964, −3.57299693726454545594161904703, −3.46193753214097762221843232993, −3.31394948679637638342325092007, −3.18914911207464435883712436734, −2.83295805592105250941081257566, −2.51655977148635525774327986799, −2.45415788013483481117057706988, −2.28328209026800020561620087986, −2.23683107475823153177732685682, −1.97388833078436812896393461025, −1.89505646226640339170563077950, −1.65962899894441374258369332099, −1.32565043292386332652646597555, −1.06959503081521705052006627166, −1.03782175870250077972096546024, −1.00815118118037333445801951871, −0.62620311924091385288034766873, −0.22370731480916464827012271222, −0.19414602813226873201606674796,
0.19414602813226873201606674796, 0.22370731480916464827012271222, 0.62620311924091385288034766873, 1.00815118118037333445801951871, 1.03782175870250077972096546024, 1.06959503081521705052006627166, 1.32565043292386332652646597555, 1.65962899894441374258369332099, 1.89505646226640339170563077950, 1.97388833078436812896393461025, 2.23683107475823153177732685682, 2.28328209026800020561620087986, 2.45415788013483481117057706988, 2.51655977148635525774327986799, 2.83295805592105250941081257566, 3.18914911207464435883712436734, 3.31394948679637638342325092007, 3.46193753214097762221843232993, 3.57299693726454545594161904703, 3.74036745745582756835684696964, 3.74898868184369707262172114265, 4.03796807114561316068123306031, 4.14192347461549325542775835977, 4.27831096655621500687976491923, 4.41718084189637908177034038363
Plot not available for L-functions of degree greater than 10.
