| L(s) = 1 | − 5-s − 2·11-s + 2·25-s + 18·29-s − 6·31-s − 14·41-s + 21·49-s + 2·55-s + 34·59-s − 6·61-s + 6·79-s + 56·89-s − 34·101-s − 19·121-s − 9·125-s + 127-s + 131-s + 137-s + 139-s − 18·145-s + 149-s + 151-s + 6·155-s + 157-s + 163-s + 167-s + 39·169-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s + 2/5·25-s + 3.34·29-s − 1.07·31-s − 2.18·41-s + 3·49-s + 0.269·55-s + 4.42·59-s − 0.768·61-s + 0.675·79-s + 5.93·89-s − 3.38·101-s − 1.72·121-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.49·145-s + 0.0819·149-s + 0.0813·151-s + 0.481·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.669234133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.669234133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T - T^{2} + 6 T^{3} - p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 3 p T^{2} + 36 p T^{4} - 2035 T^{6} + 36 p^{3} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + T + p T^{2} - 24 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 3 p T^{2} + 867 T^{4} - 13430 T^{6} + 867 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 66 T^{2} + 1995 T^{4} - 39444 T^{6} + 1995 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 15 T^{2} + 72 T^{3} + 15 p T^{4} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 45 T^{2} + 2220 T^{4} - 50283 T^{6} + 2220 p^{2} T^{8} - 45 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 3 T + p T^{2} )^{6} \) |
| 31 | \( ( 1 + 3 T + 15 T^{2} - 172 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 162 T^{2} + 12051 T^{4} - 549268 T^{6} + 12051 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 7 T + 104 T^{2} + 477 T^{3} + 104 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 27 T^{2} + 2091 T^{4} - 147790 T^{6} + 2091 p^{2} T^{8} - 27 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 177 T^{2} + 14352 T^{4} - 773787 T^{6} + 14352 p^{2} T^{8} - 177 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 54 T^{2} + 6183 T^{4} - 318004 T^{6} + 6183 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 17 T + 257 T^{2} - 2094 T^{3} + 257 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 3 T + 144 T^{2} + 397 T^{3} + 144 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 297 T^{2} + 41736 T^{4} - 3513043 T^{6} + 41736 p^{2} T^{8} - 297 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 171 T^{2} - 72 T^{3} + 171 p T^{4} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 54 T^{2} - 2049 T^{4} + 466316 T^{6} - 2049 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 3 T + 93 T^{2} - 366 T^{3} + 93 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 93 T^{2} + 18516 T^{4} - 1028715 T^{6} + 18516 p^{2} T^{8} - 93 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 28 T + 512 T^{2} - 5646 T^{3} + 512 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 243 T^{2} + 18171 T^{4} - 722050 T^{6} + 18171 p^{2} T^{8} - 243 p^{4} T^{10} + p^{6} T^{12} \) |
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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
−5.01920239658685234725158719656, −4.74623316190789654542311940349, −4.53815840055412135535226853737, −4.47645730218212514539712461318, −4.26304822690885536934897831696, −4.25648610684150837396868777163, −3.99693877907860927717910264055, −3.63866500471855697411548361755, −3.62551841075756196207728543359, −3.59182652787504053053240308299, −3.47229913130544343591230840836, −3.06601709165197626600027130063, −2.88913319805570886551325435049, −2.79038081384094175277613527694, −2.65186690949621203313024838157, −2.48353746806309346876229104173, −2.28028821726144664956568555637, −2.01030207605220562837040614207, −1.75728708931563229761306017033, −1.72257037329489208008280111921, −1.45087985496843532425502901088, −0.78083858101985060188693368648, −0.77553547349887033334375542439, −0.75316952157876241038749545202, −0.40961284485859985984533296879,
0.40961284485859985984533296879, 0.75316952157876241038749545202, 0.77553547349887033334375542439, 0.78083858101985060188693368648, 1.45087985496843532425502901088, 1.72257037329489208008280111921, 1.75728708931563229761306017033, 2.01030207605220562837040614207, 2.28028821726144664956568555637, 2.48353746806309346876229104173, 2.65186690949621203313024838157, 2.79038081384094175277613527694, 2.88913319805570886551325435049, 3.06601709165197626600027130063, 3.47229913130544343591230840836, 3.59182652787504053053240308299, 3.62551841075756196207728543359, 3.63866500471855697411548361755, 3.99693877907860927717910264055, 4.25648610684150837396868777163, 4.26304822690885536934897831696, 4.47645730218212514539712461318, 4.53815840055412135535226853737, 4.74623316190789654542311940349, 5.01920239658685234725158719656
Plot not available for L-functions of degree greater than 10.
